tag:blogger.com,1999:blog-49876091144152055932018-08-20T15:18:34.299+01:00M-PhiA blog dedicated to mathematical philosophy.Jeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger556125tag:blogger.com,1999:blog-4987609114415205593.post-56954510927030670032018-08-06T20:50:00.003+01:002018-08-06T20:50:38.285+01:00Postdoc in formal epistemology & law<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">A postdoc position (3 years, fixed term) 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 application deadline is September 15, 2018. More details <a href="https://entiaetnomina.blogspot.com/2018/08/postdoc-position-in-formal-epistemology.html">here</a>.</div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-87835110580353858692018-07-30T21:16:00.002+01:002018-07-30T21:27:18.688+01:00The Dutch Book Argument for RegularityI've just signed a contract with Cambridge University Press to write <a href="https://richardpettigrew.com/books/the-dutch-book-argument/" target="_blank">a book on the Dutch Book Argument</a> for their Elements in Decision Theory and Philosophy series. So over the next few months, I'm going to be posting some bits and pieces as I get properly immersed in the literature.<br /><br />-------<br /><br />We say that a probabilistic credence function $c : \mathcal{F} \rightarrow [0, 1]$ is <i>regular</i> if $c(A) > 0$ for all propositions $A$ in $\mathcal{F}$ such that there is some world at which $A$ is true.<br /><br /><b>The Principle of Regularity (standard version)</b> If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, rationality requires that $c$ is regular.<br /><br />I won't specify which worlds are in the scope of the quantifier over worlds that occurs in the antecedent of this norm. It might be all the logically possible worlds, or the metaphysically possible worlds, or the conceptually possible worlds; it might be the epistemically possible worlds. Different answers will give different norms. But we needn't decide the issue here. We'll just specify that it's the same set of worlds that we quantify over in the Dutch Book argument for Probabilism when we say that, if your credences aren't probabilistic, then there's a series of bets they'll lead you to enter into that will lose you money<i> at all possible worlds</i>. <br /><br />In this post, I want to consider the almost-Dutch Book Argument for the norm of Regularity. Here's how it goes: Suppose you have a credence $c(A) = 0$ in a proposition $A$, and suppose that $A$ is true at world $w$. Then, recall, the first premise of the standard Dutch Book argument for Probabilism:<br /><br /><b>Ramsey's Thesis</b> If your credence in a proposition $X$ is $c(X) = p$, then you're permitted to pay £$pS$ for a bet that returns £$S$ if $X$ is true and £$0$ if $X$ is false, for any $S$, positive or negative or zero.<br /><br />So, since $c(A) = 0$, your credences will permit you to sell the following bet for £0: if $A$, you must pay out £1; if $\overline{A}$, you will pay out £0. But selling this bet for this price is weakly dominated by refusing the bet. Selling the bet at that price loses you money in all $A$-worlds, and gains you nothing in $\overline{A}$-worlds. Whereas refusing the bet neither loses nor gains you anything in any world. Thus, your credences permit you to choose a weakly dominated act. So they are irrational. Or so the argument goes. I call this the almost-Dutch Book argument for Regularity since it doesn't punish you with a sure loss, but rather with a possible loss with no compensating possible gain.<br /><br />If this argument works, it establishes the standard version of Regularity stated above. But consider the following case. $A$ and $B$ are two logically independent propositions -- <i>It will be rainy tomorrow</i> and <i>It will be hot tomorrow</i>, for instance. You have only credences in $A$ and in the conjunction $AB$. You don't have credences in $\overline{A}$, $A \vee B$, $A\overline{B}$, and so on. What's more, your credences in $A$ and $AB$ are equal, i.e., $c(A) = c(AB)$. That is, you are exactly as confident in $A$ as you are in its conjunction with $B$. Then, in some sense, you violate Regularity, though you don't violate the standard version we stated above. After all, since your credence in $A$ is the same as your credence in $AB$, you must give no credence whatsoever to the worlds in which $A$ is true and $B$ is false. If you did, then you would set $c(AB) < c(A)$. But you don't have a credence in $A\overline{B}$. So there is no proposition true at some worlds to which you assign a credence of 0. Thus, the almost-Dutch Book argument sketched above will not work. We need a different Dutch Book argument for the following version of Regularity:<br /><br /><b>The Principle of Regularity (full version)</b> If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, then rationality requires that there is an extension $c^*$ of $c$ to a full algebra $\mathcal{F}^*$ that contains $\mathcal{F}$ such that $c^* : \mathcal{F}^* \rightarrow [0, 1]$ is regular.<br /><br />It is this principle that you violate if $c(A) = c(AB)$ when $A$ and $B$ are logically independent. For any probabilistic extension $c^*$ of $c$ that assigns a credence to $A\overline{B}$ must assign it credence 0 even though there is a world at which it is true.<br /><br />How are we to give an almost-Dutch Book argument for this version of Regularity? There are two possible approaches.<br /><br />On the first, we strengthen the first premise of the standard Dutch Book argument. Ramsey's Thesis says: if you have credence $c(X) = p$ in $X$, then you are permitted to pay £$pS$ for a bet that pays £$S$ if $X$ and £$0$ if $\overline{X}$. The stronger version says:<br /><br /><b>Strong Ramsey's Thesis</b> If every extension $c^*$ of $c$ to a full algebra $\mathcal{F}^*$ that contains $\mathcal{F}$ is such that $c^*(X) = p$, then you are permitted to pay £$pS$ for a bet that pays £$S$ if $X$ and £$0$ if $\overline{X}$.<br /><br />The idea is that, if every extension assigns the same credence $p$ to $X$, then you are in some sense committed to assigning credence $p$ to $X$. And thus, you are permitted to enter into which ever bets you'd be permitted to enter into if you actually had credence $p$.<br /><br />On the second approach to giving an almost-Dutch Book argument for the full version of the Regularity principle, we actually provide an almost-Dutch Book using just the credences that you do in fact assign. Suppose, for instance, you have credence $c(A) = c(AB) = 0.5$. Then you will sell for £5 a bet that pays out £10 if $A$ and £0 if $\overline{A}$, while you will buy for £5 a bet that pays £10 if $AB$ and £0 if $\overline{AB}$. Then, if $A$ is true and $B$ is true, you will have a net gain of £0, and similarly if $A$ is false. But if $A$ is true and $B$ is false, you will lose £10. Thus, you face the possibility of loss with no possibility of gain. Now, the question is: can we always construct such almost-Dutch Books? And the answer is that we can, as the following theorem shows:<br /><b><br /></b><b>Theorem 1 (Almost-Dutch Book Theorem for Full Regularity) </b>Suppose $\mathcal{F} = \{X_1, \ldots, X_n\}$ is a set of propositions. Suppose $c : \mathcal{F} \rightarrow [0, 1]$ is a credence function that cannot be extended to a regular probability function on a full algebra $\mathcal{F}^*$ that contains $\mathcal{F}$. Then there is a sequence of stakes $S = (S_1, \ldots, S_n)$, such that if, for each $1 \leq i \leq n$, you pay £$(c(X_i) \times S_i)$ for a bet that pays out £$S_i$ if $X_i$ and £0 if $\overline{X_i}$, then the total price you'll pay is at least the pay off of these bets at all worlds, and more than the payoff at some.<br /><br />That is,<br />(i) for all worlds $w$,<br />$$S\cdot (w - c) = S \cdot w - S \cdot c = \sum^n_{i=1} S_iw(X_i) + \sum^n_{i=1} S_ic(X_i) \leq 0$$<br />(ii) for some worlds $w$, <br />$$S\cdot (w - c) = S \cdot w - S \cdot c = \sum^n_{i=1} S_iw(X_i) + \sum^n_{i=1} S_ic(X_i) \leq 0$$<br />where $w(X_i) = 1$ if $X_i$ is true at $w$ and $w(X_i) = 0$ if $X_i$ is false at $w$. We call $w(-)$ the indicator function of $w$. <br /><br /><i>Proof sketch. </i>First, recall de Finetti's observation that your credence function $c : \mathcal{F} \rightarrow [0, 1]$ is a probability function iff it is in the convex hull of the indicator functions of the possible worlds -- that is, iff $c$ is in $\{w(-) : w \mbox{ is a possible world}\}^+$. Second, note that, if your credence function can't be extended to a regular credence function, it sits on the boundary of this convex hull. In particular, if $W_c = \{w' : c = \sum_w \lambda_w w \Rightarrow \lambda_{w'} > 0\}$, then $c$ lies on the boundary surface created by the convex hull of $W_c$. Third, by the Supporting Hyperplane Theorem, there is a vector $S$ such that $S$ is orthogonal to this boundary surface and thus:<br />(i) $S \cdot (w-c) = S \cdot w - S \cdot c = 0$ for all $w$ in $W_c$; and<br />(ii) $S \cdot (w-c) = S \cdot w - S \cdot c < 0$ for all $w$ not in $W_c$.<br />Fourth, recall that $S \cdot w$ is the total payout of the bets at world $w$ and $S \cdot c$ is the price you'll pay for it. $\Box$Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-11733654836568522122018-07-26T08:19:00.002+01:002018-07-26T11:32:59.271+01:00Dutch Strategy Theorems for Conditionalization and Superconditionalization<br />I've just signed a contract with Cambridge University Press to write <a href="https://richardpettigrew.com/books/the-dutch-book-argument/" target="_blank">a book on the Dutch Book Argument</a> for their Elements in Decision Theory and Philosophy series. So over the next few months, I'm going to be posting some bits and pieces as I get properly immersed in the literature.<br /><br />----- <br /><br />Many Bayesians formulate the update norm of Bayesian epistemology as follows:<br /><br /><b>Bayesian Conditionalization</b> If<br />(i) your credence function at $t$ is $c : \mathcal{F} \rightarrow [0, 1]$,<br />(ii) your credence function at a later time $t'$ is $c' : \mathcal{F} \rightarrow [0, 1]$,<br />(iii) $E$ is the strongest evidence you acquire between $t$ and $t'$,<br />(iv) $E$ is in $\mathcal{F}$, <br />then rationality requires that, if $c(E) > 0$, then for all $X$ in $\mathcal{F}$, $$c'(X) = c(X|E) = \frac{c(XE)}{c(E)}$$<br /><br />I don't. One reason you might fail to conditionalize between $t$ and $t'$ is that you re-evaluate the options between those times. You might disavow the prior that you had at the earlier time, perhaps decide it was too biased in one way or another, or not biased enough; perhaps you come to think that it doesn't give enough consideration to the explanatory power one hypothesis would have were it true, or gives too much consideration to the adhocness of another hypothesis; and so on. Now, it isn't irrational to change your mind. So surely it can't be irrational to fail to conditionalize as a result of changing your mind in this way. On this, I agree with van Fraassen.<br /><br />Instead, I prefer to formulate the update norm as follows -- I borrow the name from Kenny Easwaran:<br /><br /><b>Plan Conditionalization</b> If<br />(i) your credence function at $t$ is $c: \mathcal{F} \rightarrow [0, 1]$,<br />(ii) between $t$ and $t'$ you will receive evidence from the partition $\{E_1, \ldots, E_n\}$,<br />(iii) each $E_i$ is in $\mathcal{F}$ <br />(iv) at $t$, your updating plan is $c'$, so that $c'_i : \mathcal{F} \rightarrow [0, 1]$ is the credence function you will adopt if $E_i$,<br />then rationality requires that, if $c(E_i) > 0$, then for all $X$ in $\mathcal{F}$, $$c'_i(X) = c(X | E_i)$$<br /><br />I want to do two things in this post. First, I'll offer what I think is a new proof of the Dutch Strategy or Diachronic Dutch Book Theorem that justifies Plan Conditionalization (I haven't come across it elsewhere, though Ray Briggs and I used the trick at the heart of it for our accuracy dominance theorem in <a href="https://onlinelibrary.wiley.com/doi/abs/10.1111/nous.12258" target="_blank">this paper</a>). Second, I'll explore how that might help us justify other norms of updating that concern situations in which you don't come to learn any proposition with certainty. We will see that we can use the proof I give to justify the following standard constraint on updating rules: Suppose the evidence I receive between $t$ and $t'$ is not captured by any of the propositions to which I assign a credence -- that is, there is no proposition $e$ to which I assign a credence that is true at all and only the worlds at which I receive the evidence I actually receive between $t$ and $t'$. As a result, there is no proposition $e$ that I learn with certainty as a result of receiving that evidence. Nonetheless, I should update my credence function from $c$ to $c'$ in such a way that it is possible to extend my earlier credence function $c$ to a credence function $c^*$ so that: (i) $c^*$ does assign a credence to $e$, and (ii) my later credence $c'(X)$ in a proposition $X$ is the credence that this extended credence function $c^*$ assigns to $X$ conditional on me receiving evidence $e$ -- that is, $c'(X) = c^*(X | e)$. That is, I should update <i>as if</i> I had assigned a credence to $e$ at the earlier time and then updated by conditionalizing on it.<br /><br />Here's the Dutch Strategy or Diachronic Dutch Book Theorem for Plan Conditionalization:<br /><br /><b>Definition (Conditionalizing pair)</b> Suppose $c$ is a credence function and $c'$ is an updating rule defined on $\{E_1, \ldots, E_n\}$. We say that $(c, c')$ <i>is a conditionalizing pair</i> if, whenever $c(E_i) > 0$, then for all $X$, $c'_i(X) = c(X | E_i)$.<br /><br /><b>Dutch Strategy Theorem</b> Suppose $(c, c')$ is not a conditionalizing pair. Then<br />(i) there are two acts $A$ and $B$ such that $c$ prefers $A$ to $B$, and<br />(ii) for each $E_i$, there are two acts $A_i$ and $B_i$ such that $c'_i$ prefers $A_i$ to $B_i$,<br />and, for each $E_i$, $A + A_i$ has greater utility than $B + B_i$ at all worlds at which $E_i$ is true.<br /><br />We'll now give the proof of this.<br /><br />First, we describe a way of representing pairs $(c, c')$. Both $c$ and each $c'_i$ are defined on the same set $\mathcal{F} = \{X_1, \ldots, X_m\}$. So we can represent $c$ by the vector $(c(X_1), \ldots, c(X_m))$ in $[0, 1]^m$, and we can represent each $c'_i$ by the vector $(c'_i(X_1), \ldots, c'_i(X_m))$ in $[0, 1]^m$. And we can represent $(c, c')$ by concatenating all of these representations to give:<br />$$(c, c') = c \frown c'_1 \frown c'_2 \frown \ldots \frown c'_n$$<br />which is a vector in $[0, 1]^{m(n+1)}$.<br /><br />Second, we use this representation to give an alternative characterization of conditionalizing pairs. First, three pieces of notation:<br /><ul><li>Let $W$ be the set of all possible worlds. </li><li>For any $w$ in $W$, abuse notation and write $w$ also for the credence function on $\mathcal{F}$ such that $w(X) = 1$ if $X$ is true at $w$, and $w(X) = 0$ if $X$ is false at $w$.</li><li>For any $w$ in $W$, let $$(c, c')_w = w \frown c'_1 \frown \ldots \frown c'_{i-1} \frown w \frown c'_{i+1} \frown \ldots \frown c'_n$$ where $E_i$ is the element of the partition that is true at $w$.</li></ul><b>Lemma 1</b> If $(c, c')$ is not a conditionalizing pair, then $(c, c')$ is not in the convex hull of $\{(c, c')_w : w \in W\}$, which we write $\{(c, c')_w : w \in W\}^+$.<br /><br /><i>Proof of Lemma 1. </i>If $(c, c')$ is in $\{(c, c')_w : w \in W\}^+$, then there are $\lambda_w \geq 0$ such that<br /><br />(1) $\sum_{w \in W} \lambda_w = 1$,<br />(2) $c(X) = \sum_{w \in W} \lambda_w w(X)$<br />(3) $c'_i(X) = \sum_{w \in E_i} \lambda_w w(X) + \sum_{w \not \in E_i} \lambda_w c'_i(X)$.<br /><br />By (2), we have $\lambda_w = c(w)$. So by (3), we have $$c'_i(X) = c(XE_i) + (1-c(E_i))c'_i(X)$$ So, if $c(E_i) > 0$, then $c'_i(X) = c(X | E_i)$.<br /><br />Third, we use this alternative characterization of conditionalizing pairs to specify the acts in question. Suppose $(c, c')$ is not a conditionalizing pair. Then $(c, c')$ is outside $\{(c, c')_w : w \in W\}^+$. Now, let $(p, p')$ be the orthogonal projection of $(c, c')$ into $\{(c, c')_w : w \in W\}^+$. Then let $(S, S') = (c, c') - (p, p')$. That is, $S = c - p$ and $S'_i = c'_i - p'_i$. Now pick $w$ in $W$. Then the angle between $(S, S')$ and $(c, c')_w - (c, c')$ is obtuse and thus<br />$$(S, S') \cdot ((c, c')_w - (c, c')) = -\varepsilon_w < 0$$<br /><br />Thus, define the acts $A$, $B$, $A'_i$ and $B'_i$ as follows:<br /><ul><li>The utility of $A$ at $w$ is $S \cdot (w - c) + \frac{1}{3}\varepsilon_w$:</li><li>The utility of $B$ at $w$ is 0;</li><li>The utility of $A'_i$ at $w$ is $S'_i \cdot (w - c'_i) + \frac{1}{3}\varepsilon_w$;</li><li>The utility of $B'_i$ at $w$ is 0.</li></ul> Then the expected utility of $A$ by the lights of $c$ is $\sum^w c(w)\frac{1}{3}\varepsilon_w > 0$, while the expected utility of $B$ is 0, so $c$ prefers $A$ to $B$. And the expected utility of $A'_i$ by the lights of $c'_i$ is $\sum_w c'_i(w)\frac{1}{3}\varepsilon_w > 0$, while the expected utility of $B'_i$ is 0, so $c'_i$ prefers $A'_i$ to $B'_i$. But the utility of $A + A'_i$ at $w$ is<br />$$S \cdot (w - c) + S'_i \cdot (w - c'_i) + \frac{2}{3}\varepsilon_w = (S, S') \cdot ((c, c')_w - (c, c')) + \frac{2}{3}\varepsilon_w = - \frac{1}{3}\varepsilon_w < 0$$<br />where $E_i$ is true at $w$. While the utility of $B + B'_i$ at $w$ is 0.<br /><br />This completes our proof. $\Box$<br /><br />You might be forgiven for wondering why we are bothering to give an alternative proof for a theorem that is already well-known. David Lewis proved the Dutch Strategy Theorem in a handout for a seminar at Princeton in 1972, Paul Teller then reproduced it (with full permission and acknowledgment) in a paper in 1973, and Lewis finally published his handout in 1997 in his collected works. Why offer a new proof?<br /><br />It turns out that this style of proof is actually a little more powerful. To see why, it's worth comparing it to an alternative proof of the Dutch Book Theorem for Probabilism, which I described in <a href="http://m-phi.blogspot.com/2013/09/the-mathematics-of-dutch-book-arguments.html" target="_blank">this post</a> (it's not original to me, though I'm afraid I can't remember where I first saw it!). In the standard Dutch Book Theorem for Probabilism, we work through each of the axioms of the probability calculus, and say how you would Dutch Book an agent who violates it. The axioms are: Normalization, which says that $c(\top) = 1$ and $c(\bot) = 0$; and Additivity, which says that $c(A \vee B) = c(A) + c(B) - c(AB)$. But consider an agent with credences only in the propositions $\top$, $A$, and $A\ \&\ B$. Her credences are: $c(\top) = 1$, $c(A) = 0.4$, $c(A\ \&\ B) = 0.7$. Then there is no axiom of the probability calculus that she violates. And thus the standard proof of the Dutch Book Theorem is no help in identifying any Dutch Book against her. Yet she is Dutch Bookable. And she violates a more expansive formulation of Probabilism that says, not only are you irrational if your credence function is not a probability function, but also if your credence function <i>cannot be extended to a probability function</i>. So the standard proof of the Dutch Book Theorem can't establish this more expansive version. But the alternative proof I mentioned above can.<br /><br />Now, something similar is true of the alternative proof of the Dutch Strategy Theorem that I offered above (I happened upon this while discussing Superconditionalizing with Jason Konek, who uses similar techniques in his argument for J-Kon, the alternative to Jeffrey's Probability Kinematics that he proposes in his paper, <a href="https://philpapers.org/rec/KONTAO-2" target="_blank">'The Art of Learning'</a>, which was runner-up for last year's Sander's Prize in Epistemology). In Lewis' proof of that theorem: First, if you violate Plan Conditionalization, there must be $E_i$ and $X$ such that $c(E_i) > 0$ and $c'_i(X) \neq c(X|E_i)$. Then you place bets on $XE_i$, $\overline{E_i}$ at the earlier time $t$, and a bet on $X$ at $t'$. These bets then together lose you money in any world at which $E_i$ is true. Now, it might seem that you must have the required credences to make those bets just in virtue of violating Plan Conditionalization. But imagine the following is true of you: between $t$ and $t'$, you'll obtain evidence from the partition $\{E_1, \ldots, E_n\}$. And, at $t'$, you'll update on this evidence using the rule $c'$. That is, if $E_i$, then you'll adopt the new credence function $c'_i$ at time $t'$. Now, you don't assign credences to the propositions in $\{E_1, \ldots, E_n\}$. Perhaps this is because you don't have the conceptual resources to formulate these propositions. So while you will update using the rule $c'$, this is not a rule you consciously or explicitly adopt, since to state it would require you to use the propositions in $\{E_1, \ldots, E_n\}$. So it's more like you have a disposition to update in this way. Now, how might we state Plan Conditionalization for such an agent? We can't demand that $c'_i(X) = c(X|E_i)$, since $c(X | E_i)$ is not defined. Rather, we demand that there is some extension $c^*$ of $c$ to a set of propositions that does include each $E_i$ such that $c'_i(X) = c^*(X | E_i)$. Thus, we have:<br /><br /><b>Plan Superconditionalization</b> If<br />(i) your credence function at $t$ is $c : \mathcal{F} \rightarrow [0, 1]$,<br />(ii) between $t$ and $t'$ you will receive evidence from the partition $\{E_1, \ldots, E_n\}$,<br />(iii) at $t$, your updating plan is $c'$, so that $c'_i : \mathcal{F} \rightarrow [0, 1]$ is the credence function you plan to adopt if $E_i$,<br />then rationality requires that there is some extension $c^*$ of $c$ for which, if $c^*(E_i) > 0$, then for all $X$, $$c'_i(X) = c^*(X | E_i)$$<br /><br />And it turns out that we can adapt the proof above for this purpose. Say that $(c, c')$ is a superconditionalizing pair if there is an extension $c^*$ of $c$ such that, if $c^*(E_i) > 0$, then for all $X$, $c'_i(X) = c^*(X | E_i)$. Then we can prove that if $(c, c')$ is not a superconditionalizing pair, then $(c, c')$ is not in $\{(c, c')_w : w \in W\}^+$. Here's the proof from above adapted to our case: If $(c, c')$ is in $\{(c, c')_w : w \in W\}^+$, then there are $\lambda_w \geq 0$ such that<br /><br />(1) $\sum_{w \in W} \lambda_w = 1$,<br />(2) $c(X) = \sum_{w \in W} \lambda_w w(X)$<br />(3) $c'_i(X) = \sum_{w \in E_i} \lambda_w w(X) + \sum_{w \not \in E_i} \lambda_w c'_i(X)$.<br /><br />Define the following extension $c^*$ of $c$: $c^*(w) = \lambda_w$. Then, by (3), we have $$c'_i(X) = c^*(XE_i) + (1-c^*(E_i))c'_i(X)$$ So, if $c^*(E_i) > 0$, then $c'_i(X) = c^*(X | E_i)$, as required. $\Box$<br /><br />Now, this is a reasonably powerful version of conditionalization. For instance, as Skyrms showed <a href="http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199652808.001.0001/acprof-9780199652808-chapter-7" target="_blank">here</a>, if we make one or two further assumptions on the extension of $c$ to $c^*$, we can derive Richard Jeffrey's Probability Kinematics from Plan Superconditionalization. That is, if the evidence $E_i$ will lead you to set your new credences across the partition $\{B_1, \ldots, B_k\}$ to $q_1, \ldots, q_k$, respectively, so that $c'_i(B_j) = q_j$, then your new credence $c'_i(X)$ must be $\sum^k_{j=1} c(X | B_j)q_j$, as Probability Kinematics demands. Thus, Plan Superconditionalization places a powerful constraint on updating rules for situations in which the proposition stating your evidence is not one to which you assign a credence. Other cases of this sort include the Judy Benjamin problem and the many cases in which MaxEnt is applied.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com3tag:blogger.com,1999:blog-4987609114415205593.post-75607084411413354532018-07-25T12:19:00.001+01:002018-07-25T12:19:05.435+01:00Deadline for PhD position in formal epistemology & law extended<div dir="ltr" style="text-align: left;" trbidi="on"><a href="http://entiaetnomina.blogspot.com/2018/04/one-phd-position-in-formal-epistemology.html">This position</a> is still available. Deadline extended to September 7, 2018.</div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-39135622723093007182018-07-25T11:06:00.002+01:002018-07-26T08:24:10.197+01:00On the Expected Utility Objection to the Dutch Book Argument for Probabilism<br />I've just signed a contract with Cambridge University Press to write <a href="https://richardpettigrew.com/books/the-dutch-book-argument/" target="_blank">a book on the Dutch Book Argument</a> for their Elements in Decision Theory and Philosophy series. So over the next few months, I'm going to be posting some bits and pieces as I get properly immersed in the literature. The following came up while thinking about Brian Hedden's paper <a href="https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-0068.2011.00842.x" target="_blank">'Incoherence without Exploitability'.</a><br /><br /><h2>What is Probabilism? </h2><br />Probabilism says that your credences should obey the axioms of the probability calculus. Suppose $\mathcal{F}$ is the algebra of propositions to which you assign a credence. Then we let $0$ represent the lowest possible credence you can assign, and we let $1$ represent the highest possible credence you can assign. We then represent your credences by your credence function $c : \mathcal{F} \rightarrow [0, 1]$, where, for each $A$ in $\mathcal{F}$, $c(A)$ is your credence in $A$.<br /><br /><b>Probabilism</b><br />If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, then rationality requires that:<br />(P1a) $c(\bot) = 0$, where $\bot$ is a necessarily false proposition;<br />(P1b) $c(\top) = 1$, where $\top$ is a necessarily true proposition;<br />(P2) $c(A \vee B) = c(A) + c(B)$, for any mutually exclusive propositions $A$ and $B$ in $\mathcal{F}$.<br /><br />This is equivalent to:<br /><br /><b>Partition Probabilism</b><br />If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, then rationality requires that, for any two partitions $\mathcal{X} = \{X_1, \ldots, X_m\}$ and $\mathcal{Y} = \{Y_1, \ldots, Y_n\}$,$$\sum^m_{i=1} c(X_i) = 1= \sum^n_{j=1} c(Y_j)$$<br /><br /><h2>The Dutch Book Argument for Probabilism</h2><br />The Dutch Book Argument for Probabilism has three premises. The first, which I will call <i>Ramsey's Thesis</i> and abbreviate <i>RT</i>, posits a connection between your credence in a proposition and the prices you are rationally permitted or rationally required to pay for a bet on that proposition. The second, known as the <i>Dutch Book Theorem</i>, establishes that, if you violate Probabilism, there is a set of bets you might face, each with a price attached, such that (i) by Ramsey's Thesis, for each bet, you are rationally required to pay the attached price for it, but (ii) the sum of the prices of the bets exceeds the highest possible payout of the bets, so that, having paid each of those prices, you are guaranteed to lose money. The third premise, which we might call the <i>Domination Thesis</i>, says that credences are irrational if they mandate you to make a series of decisions (i.e, paying certain prices for the bets) that is guaranteed to leave you worse off than another series of decisions (i.e., refusing to pay those prices for the bets)---in the language of decision theory, paying the attached price for each of the bets is<i> dominated</i> by refusing each of the bets, and credences that mandate you to choose dominated options are irrational. The conclusion of the Dutch Book Argument is then Probabilism. Thus, the argument runs:<br /><br /><b>The Dutch Book Argument for Probabilism</b><br />(DBA1) Ramsey's Thesis<br />(DBA2) Dutch Book Theorem<br />(DBA3) Domination Thesis<br />Therefore,<br />(DBAC) Probabilism<br /><br />The argument is valid. The second premise is a mathematical theorem. Thus, if the argument fails, it must be because the first or third premise is false, or both. In this paper, we focus on the first premise, and the expected utility objection to it. So, let's set out that premise in a little more detail.<br /><br />In what follows, we assume that (i) you are risk-neutral, and (ii) that there is some quantity such that your utility is linear in that quantity---indeed, we will speak as if your utility is linear in money, but that is just for ease of notation and familiarity; any quantity would do. Neither (i) nor (ii) is realistic, and indeed these idealisations are the source of other objections to Ramsey's Thesis. But they are not our concern here, so we will grant them.<br /><br /><b>Ramsey's Thesis (RT)</b> Suppose your credence in $X$ in $c(X)$. Consider a bet that pays you £$S$ if $X$ is true and £0 if $X$ is false, where $S$ is a real number, either positive, negative, or zero---$S$ is called the <i>stake</i> of the bet. You are offered this bet for the price £$x$, where again $x$ is a real number, either positive, negative, or zero. Then:<br />(i) If $x < c(X) \times S$, you are rationally required to pay £$x$ to enter into this bet;<br />(ii) If $x = c(X) \times S$, you are rationally permitted to pay £$x$ and rationally permitted to refuse;<br />(iii) If $x > c(X) \times S$, you are rationally required to refuse.<br /><br />Roughly speaking, Ramsey's Thesis says that, the more confident you are in a proposition, the more you should be prepared to pay for a bet on it. More precisely, it says: (a) if you have minimal confidence in that proposition (i.e. 0), then you should be prepared to pay nothing for it; (b) if you have maximal confidence in it (i.e. 1), then you should be prepared to pay the full stake for it; (c) for levels of confidence in between, the amount you should be prepared to pay increases linearly with your credence.<br /><br /><h2>The Expected Utility Objection</h2><br />We turn now to the objection to Ramsey's Thesis (RT) we wish to treat here. Hedden (2013) begins by pointing out that we have a general theory of how credences and utilities should guide action: <br /><br /><blockquote class="tr_bq">Given a set of options available to you, expected utility theory says that your credences license you to choose the option with the highest expected utility, defined as:</blockquote><blockquote>$$\mathrm{EU}(A) = \sum_i P(O_i|A) \times U(O_i)$$<br />On this view, we should evaluate which bets your credences license you to accept by looking at the expected utilities of those bets. (Hedden, 2013, 485)</blockquote><br />He considers the objection that this only applies when credences satisfy Probabilism, but rejects it:<br /><br /><blockquote class="tr_bq">In general, we should judge actions by taking the sum of the values of each possible outcome of that action, weighted by one's credence that the action will result in that outcome. This is a very intuitive proposal for how to evaluate actions that applies even in the context of incoherent credences. (Hedden, 2013, 486)</blockquote><br />Thus, Hedden contends that we should always choose by maximising expected utility relative to our credences, whether or not those credences are coherent. Let's call this principle <i>Maximise Subjective Expected Utility</i> and abbreviate it <i>MSEU</i>. He then observes that MSEU conflicts with RT. Consider, for instance, Cináed, who is 60% confident it will rain and 20% confident it won't. According to RT, he is rationally required to sell for £65 a bet in which he pays out £100 if it rains and £0 if is doesn't. But the expected utility of this bet for him is$$0.6 \times (-100 + 65) + 0.2 \times (-0 + 65) = -8$$That is, it has lower expected utility than refusing to sell the bet, since his expected utility for doing that is$$0.6 \times 0 + 0.2 \times 0 = 0$$So, while RT says you must sell that bet for that price, MSEU says you must not. So RT and MSEU are incompatible, and Hedden claims that we should favour MSEU. There are two ways to respond to this. On the first, we try to retain RT in some form in spite of Hedden's objection---I call this the <i>permissive response</i> below. On the second, we try to give a pragmatic argument for Probabilism using MSEU instead of RT---I call this the <i>bookless response</i> below. In the following sections, I will consider these in turn.<br /><br /><h2>The Permissive Response</h2><br />While Hedden is right to say that maximising expected utility in line with Maximise Subjective Expected Utility (MSEU) is intuitively rational even when your credences are incoherent, so is Ramsey's Thesis (RT). It is certainly intuitively correct that, to quote Hedden, ''we should judge actions by taking the sum of the values of each possible outcome of that action, weighted by one's credence that the action will result in that outcome.'' But it is also intuitively correct that, to quote from our gloss of Ramsey's Thesis above, ''(a) if you have minimal confidence in that proposition (i.e. 0), then you should be prepared to pay nothing for it; (b) if you have maximal confidence in it (i.e. 1), then you should be prepared to pay the full stake for it; (c) for levels of confidence in between, the amount you should be prepared to pay increases linearly with your credence.'' What are we to do in the face of this conflict between our intuitions?<br /><br />One natural response is to say that choosing in line with RT is rationally permissible and choosing in line with MSEU is also rationally permissible. When your credences are coherent, the dictates of MSEU and RT are the same. But when you are incoherent, they are sometimes different, and in that situation you are allowed to follow either. In particular, faced with a bet and proposed price, you are permitted to pay that price if it is permitted by RT <i>and</i> you are permitted to pay it if it is permitted by MSEU.<br /><br />If this is right, then we can resurrect the Dutch Book Argument with a permissive version of RT as the first premise:<br /><br /><b>Permissive Ramsey's Thesis</b> Suppose your credence in $X$ in $c(X)$. Consider a bet that pays you £$S$ if $X$ is true and £0 if $X$ is false. You are offered this bet for the price £$x$. Then:<br />(i) If $x \leq c(X) \times S$, you are rationally permitted to pay £$x$ to enter into this bet.<br /><br />And we could then amend the third premise---the Domination Thesis (DBA3)---to ensure we could still derive our conclusion. Instead of saying that credences are irrational if they <i>mandate</i> you to make a series of decisions that is guaranteed to leave you worse off than another series of decisions, we might say that credences are irrational if they <i>permit</i> you to make a series of decisions that is guaranteed to leave you worse off than another series of decisions. In the language of decision theory, instead of saying only that credences that <i>mandate</i> you to choose dominated options are irrational, we say also that credences that <i>permit</i> you to choose dominated options are irrational. We might call this the <i>Permissive Domination Thesis</i>.<br /><br />Now, by weakening the first premise in this way, we respond to Hedden's objection and make the premise more plausible. But we strengthen the third premise to compensate and perhaps thereby make it less plausible. However, I imagine that anyone who accepts one of the versions of the third premise---either the Domination Thesis or the Permissive Domination Thesis---will also accept the other. Having credences that <i>mandate</i> dominated choices may be worse than having credences that <i>permit</i> such choices, but both seem sufficient for irrationality. Perhaps the former makes you <i>more</i> irrational than the latter, but it seems clear that the ideally rational agent will have credences that do neither. And if that's the case, then we can replace the standard Dutch Book Argument with a slight modification:<br /><br /><b>The Permissive Dutch Book Argument for Probabilism</b><br />(PDBA1) Permissive Ramsey's Thesis<br />(PDBA2) Dutch Book Theorem<br />(PDBA3) Permissive Domination Thesis<br />Therefore,<br />(PDBAC) Probabilism<br /><br /><h2>The Bookless Response</h2><br />Suppose you refuse even the permissive version of RT, and insist that coherent and incoherent agents alike should choose in line with MSEU. Then what becomes of the Dutch Book Argument? As we noted above, Hedden shows that it fails---MSEU is not sufficient to establish the conclusion. In particular, Hedden gives an example of an incoherent credence function that is not Dutch Bookable via MSEU. That is, there are no sets of bets with accompanying prices such that (a) MSEU will demand that you pay each of those prices, and (b) the sum of those prices is guaranteed to exceed the sum of the payouts of that set of bets. However, as we will see, accepting individual members of such a set of bets is just one way to make bad decisions based on your credences.<br /><br />Consider Hedden's example. In it, you assign credences to propositions in the algebra built up from three possible worlds, $w_1$, $w_2$, and $w_3$. Here are some of your credences:<br /><ul><li>$c(w_1 \vee w_2) = 0.8$ and $c(w_3) = 0$</li><li>$c(w_1) = 0.7$ and $c(w_2 \vee w_3) = 0$</li></ul>Now, consider the following two options, $A$ and $B$, whose utilities in each state of the world are set out in the following table:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-ekyYtB-ehnw/W1hJ6FuFhHI/AAAAAAAAAvE/1lIqibrpdTsF3vuG8g6RYm2WBpHBgmMewCLcBGAs/s1600/IMG_6046.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="494" data-original-width="1016" height="96" src="https://4.bp.blogspot.com/-ekyYtB-ehnw/W1hJ6FuFhHI/AAAAAAAAAvE/1lIqibrpdTsF3vuG8g6RYm2WBpHBgmMewCLcBGAs/s200/IMG_6046.jpg" width="200" /></a></div><br />Then notice first that $A$ dominates $B$---that is, the utility of $A$ is higher than $B$ in every possible state of the world. But, using your incoherent credences, you assign a higher expected utility to $B$ than to $A$. Your expected utility for $A$---which must be calculated relative to your credences in $w_1$ and $w_2 \vee w_3$, since the utility of $A$ given $w_1 \vee w_2$ is undefined---is $0.7 \times 78 + 0 \times 77 = 54.6$. And your expected utility for $B$---which must be calculated relative to your credences in $w_1 \vee w_2$ and $w_3$, since the utility of $B$ given $w_2 \vee w_3$ is undefined---is $0.8 \times 74 + 0 \times 75 = 59.2$. So, while Hedden might be right that MSEU won't leave you vulnerable to a Dutch Book, it will leave you vulnerable to choosing a dominated option. And since what is bad about entering a Dutch Book is that it is a dominated option---it is dominated by the option of refusing the bets---the invulnerability to Dutch Books should be no comfort to you.<br /><br />Now, this raises the question: For which incoherence credences is it guaranteed that MSEU won't lead you to choose a dominated option? Is it <i>all</i> incoherent credences, in which case we would have a new Dutch Book Argument for Probabilism from MSEU rather than RT? Or is it some subset? Below, we prove a theorem that answers that. First, a weakened version of Probabilism:<br /><br /><b>Bounded Probabilism</b> If $c : \mathcal{F}\rightarrow [0, 1]$ is your credence function, then rationality requires that:<br />(BP1a) $c(\bot) = 0$, where $\bot$ is a necessarily false proposition;<br />(BP1b) There is $0 < M \leq 1$ such that $c(\top) = M$, where $\top$ is a necessarily true proposition;<br />(BP2) $c(A \vee B) = c(A) + c(B)$, if $A$ and $B$ are mutually exclusive.<br /><br />Bounded Probabilism says that you should have lowest possible credence in necessary falsehoods, some positive credence---not necessarily 1---in necessary truths, and your credence in a disjunction of two incompatible propositions should be the sum of your credences in the disjuncts.<br /><br /><b>Theorem 1</b> The following are equivalent:<br />(i) $c$ satisfies Bounded Probabilism<br />(ii) For all options $A$, $B$, if $A$ dominates $B$, then $\mathrm{EU}_c(A) > \mathrm{EU}_c(B)$.<br /><br />The proof is in the Appendix below. Thus, even without Ramsey's Thesis or the permissive version described above, you can still give a pragmatic argument for a norm that lies very close to Probabilism, namely, Bounded Probabilism. On its own, this argument cannot say what is wrong with someone who gives less than the highest possible credence to necessary truths, but it does establish the other requirements that Probabilism imposes. To see just how close to Probabilism lies Bounded Probabilism, consider the following two norms, which are equivalent to it:<br /><br /><b>Scaled Probabilism</b> If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, then rationality requires that there is $0 < M \leq 1$ and a probability function $p : \mathcal{F} \rightarrow [0, 1]$ such that $c(-) = M \times p(-)$.<br /><br /><b>Bounded Partition Probabilism</b> If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, then rationality requires that, for any two partitions $\mathcal{X} = \{X_1, \ldots, X_m\}$ and $\mathcal{Y} = \{Y_1, \ldots, Y_n\}$,$$\sum^m_{i=1} c(X_i) = \sum^n_{j=1} c(Y_j)<br />$$Then<br /><br /><b>Lemma 2</b> The following are equivalent:<br />(i) Bounded Probabilism<br />(ii) Scaled Probabilism<br />(iii) Bounded Partition Probabilism<br /><br />As before, the proof is in the Appendix.<br /><br />So, on its own, MSEU can deliver us very close to Probabilism. But it cannot establish (P1b), namely, $c(\top) = 1$. However, I think we can also appeal to a highly restricted version of the Permissive Ramsey's Thesis to secure (P1b) and push us all the way to Probabilism.<br /><br />Consider Dima and Esther. They both have minimal confidence---i.e. 0---that it won't rain tomorrow. But Dima has credence 0.01 that it will rain, while Esther has credence 0.99 that it will. If we permit only actions that maximise expected utility, then Dima and Esther are required to pay exactly the same prices for bets on rain---that is, Dima will be required to pay a price exactly when Esther is. After all, if £$S$ is the payoff when it rains, £0 is the payoff when it doesn't, and $x$ is a proposed price, then $0.01\times (S- x) + 0 \times (0-x) \geq 0$ iff $0.99 \times (S-x) + 0 \times (0-x) \geq 0$ iff $S \geq x$. So, according to MSEU, Dima and Esther are rationally required to pay anything up to the stake of the bet for such a bet. But this is surely wrong. It is surely at least permissible for Dima to refuse to pay a price that Esther accepts. It is surely permissible for Esther to pay £99 for a bet on rain that pays £100 if it rains and £0 if it doesn't, while Dima refuses to pay anything more than £1 for such a bet, in line with Ramsey's Thesis. Suppose Dima were offered such a bet for the price of £99, and suppose she then defended her refusal to pay that price saying, 'Well, I only think it's 1% likely to rain, so I don't want to risk such a great loss with so little possible gain when I think the gain is so unlikely'. Then surely we would accept that as a rational defence.<br /><br />In response to this, defenders of MSEU might concede that RT is sometimes the correct norm of action when you are incoherent, but only in very specific cases, namely, those in which you have a positive credence in a proposition, minimal credence (i.e. 0) in its negation, and you are considering the price you might pay for a bet on that proposition. In all other cases---that is, in any case in which your credences in the proposition and its negation are both positive, or in which you are considering an action other than a bet on a proposition---you should use MSEU. I have some sympathy with this. But, fortunately, this restricted version is all we need. After all, it is precisely by applying Ramsey's Thesis to such a case that we can produce a Dutch Book against someone with $c(\bot) = 0$ and $c(\top) < 1$---we simply offer to pay them £$c(\top) \times 100$ for a bet in which they will pay out £100 if $\top$ is true and £0 if it is false; this is then guaranteed to lose them £$100 \times (1-c(X))$, which is positive. Thus, we end up with a disjunctive pragmatic argument for Probabilism: if $c(\bot) = 0$ and $c(\top) < 1$, then RT applies and we can produce a Dutch Book against you; if you violate Probabilism in any other way, then you violate Bounded Probabilism and we can then produce two options $A$ and $B$ such that $A$ dominates $B$, but your credences, via MSEU, dictate that you should choose $B$ over $A$. This, then, is our bookless pragmatic argument for Probabilism:<br /><br /><b>Bookless Pragmatic Argument for Probabilism</b><br />(BPA1) If $c$ violates Probabilism, then either (i) $c(\bot) = 0$ and $c(\top) < 1$, or (ii) $c$ violates Bounded Probabilism.<br />(BPA2) If $c(\bot) = 0$ and $c(\top) < 1$, then RT applies, and there is a bet on $\top$ such that you are required by RT to pay a higher price for that bet than its guaranteed payoff. Thus, there are options $A$ and $B$ (namely, <i>refuse the bets</i> and <i>pay the price</i>), such that $A$ dominates $B$, but RT demands that you choose $B$ over $A$.<br />(BPA3) If $c$ violates Bounded Probabilism, then by Theorem 1, there are options $A$ and $B$ such that $A$ dominates $B$, but RT demands that you choose $B$ over $A$. Therefore, by (BPA1), (BPA2), and (BPA3),<br />(BPA4) If $c$ violates Probabilism, then there are options $A$ and $B$ such that $A$ dominates $B$, but rationality requires you to choose $B$ over $A$.<br />(BPA5) Dominance Thesis<br />Therefore,<br />(BPAC) Probabilism<br /><br /><h2>Conclusion</h2><br />The Dutch Book Argument for Probabilism assumes Ramsey's Thesis, which determines the prices an agent is rationally required to pay for a bet. Hedden argues that Ramsey's Thesis is wrong. He claims that Maximise Subjective Expected Utility determines those prices, and it often disagrees with RT. In our Permissive Dutch Book Argument, I suggested that, in the face of that disagreement, we might be permissive: agents are permitted to pay any price that is required or permitted by RT and they are permitted to pay any price that is required or permitted by MSEU. In our Bookless Pragmatic Argument, I then explored what we might do if we reject this permissive response and insist that only prices permitted or required by MSEU are permissible. I showed that, in that case, we can give a pragmatic argument for Bounded Probabilism, which comes close to Probabilism, but doesn't quite reach; and I showed that, if we allow RT in the very particular cases in which it agrees better with intuition than MSEU does, we can give a pragmatic argument for Probabilism.<br /><br /><h2>Appendix: Proof of Theorem 1</h2><br /><b>Theorem 1 </b>The following are equivalent:<br />(i) $c$ satisfies Bounded Probabilism<br />(ii) For all options $A$, $B$, if $A$ dominates $B$, then $\mathrm{EU}_c(A) > \mathrm{EU}_c(B)$.<br /><br />($\Rightarrow$) Suppose $c$ satisfies Bounded Probabilism. Then, by Lemma 2, there is $0 < M \leq 1$ and a probability function $p$ such that $c(-) = M \times p(-)$. Now suppose $A$ and $B$ are actions. Then<br /><ul><li>$\mathrm{EU}_c(A) = \mathrm{EU}_{M \times p}(A) = M \times \mathrm{EU}_p(A)$</li><li>$\mathrm{EU}_c(B) = \mathrm{EU}_{M \times p}(B) = M \times \mathrm{EU}_p(B)$</li></ul>Thus, $\mathrm{EU}_c(A) > \mathrm{EU}_c(B)$ iff $\mathrm{EU}_p(A) > \mathrm{EU}_p(B)$. And we know that, if $A$ dominates $B$ and $p$ is a probability function, then $\mathrm{EU}_p(A) > \mathrm{EU}_p(B)$.<br /><br />($\Leftarrow$) Suppose $c$ violates Bounded Probabilism. Then there are partitions $\mathcal{X} = \{X_1, \ldots, X_m\}$ and $\mathcal{Y} = \{Y_1, \ldots, Y_n\}$ such that $$\sum^m_{i=1} c(X_i) = x < y = \sum^n_{j=1} c(Y_j)$$We will now define two acts $A$ and $B$ such that $A$ dominates $B$, but $\mathrm{EU}_c(A) < \mathrm{EU}_c(B)$.<br /><ul><li>For any $X_i$ in $\mathcal{X}$, $$U(A, X_i) = y - i\frac{y-x}{2(m + 1)}$$</li><li>For any $Y_j$ in $\mathcal{Y}$,$$U(B, Y_j) = x + j\frac{y-x}{2(n + 1)}$$</li></ul>Then the crucial facts are:<br /><ul><li>For any two $X_i \neq X_j$ in $\mathcal{X}$,$$U(A, X_i) \neq U(A, X_j)$$</li><li>For any two $Y_i \neq Y_j$ in $\mathcal{Y}$,$$U(B, Y_i) \neq U(B, Y_j)$$</li><li>For any $X_i$ in $\mathcal{X}$ and $Y_j$ in $\mathcal{Y}$, $$x < U(B, Y_j) < \frac{x+y}{2} < U(A, X_i) < y$$</li></ul>So $A$ dominates $B$, but$$\mathrm{EU}_c(A) = \sum^m_{i=1} c(X_i) U(A, X_i) < \sum^m_{i=1} c(X_i) \times y = xy$$<br />while$$\mathrm{EU}_c(B) = \sum^n_{j=1} c(Y_i) U(B, Y_j) > \sum^n_{j=1} c(Y_j) \times x = yx$$So $\mathrm{EU}_c(B) > \mathrm{EU}_c(A)$, as required.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-13476030767335627332018-07-19T09:26:00.001+01:002018-07-26T08:24:39.726+01:00What is Probabilism?<br />I've just signed a contract with Cambridge University Press to write <a href="https://richardpettigrew.com/books/the-dutch-book-argument/" target="_blank">a book on the Dutch Book Argument</a> for their Elements in Decision Theory and Philosophy series. So over the next few months, I'm going to be posting some bits and pieces as I get properly immersed in the literature.<br /><br />---- <br /><br />Probabilism is the claim that your credences should satisfy the axioms of the probability calculus. Here is an attempt to state the norm more precisely, where $\mathcal{F}$ is the algebra of propositions to which you assign credences and $c$ is your credence function, which is defined on $\mathcal{F}$, so that $c(A)$ is your credence in $A$, for each $A$ in $\mathcal{F}$.<br /><br /><b>Probabilism (initial formulation)</b> <br /><ul><li>(Non-Negativity) Your credences should not be negative. In symbols: $c(A) \geq 0$, for all $A$ in $\mathcal{F}$.</li><li>(Normalization I) Your credence in a necessarily false proposition should be 0. In symbols: $c(\bot) = 0$.</li><li>(Normalization II) Your credence in a necessarily true proposition should be 1. In symbols: $c(\top) = 1$.</li><li>(Finite Additivity) Your credence in the disjunction of two mutually exclusive propositions should be the sum of your credences in the disjuncts. In symbols: $c(A \vee B) = c(A) + c(B)$.</li></ul>This sort of formulation is fairly typical. But I think it's misleading in various ways.<br /><br />As is often pointed out, 0 and 1 are merely conventional choices. Like utilities, we can measure credences on different scales. But what are they conventional choices for? It seems to me that they must represent the lowest possible credence you can have and the highest possible credence you can have, respectively. After all, what we want Normalization I and II to say is that we should have lowest possible credence in necessary falsehoods and highest possible credence in necessary truths. It follows that Non-Negativity is not a normative constraint on your credences, which is how it is often presented. Rather, it follows immediately from the particular representation of our credences that we have chosen to. Suppose we chose a different representation, where -1 represents the lowest possible credence and 1 represents the highest. Then Normalization I and II would say that $c(\bot) = -1$ and $c(\top) = 1$, so Non-Negativity would be false.<br /><br />One upshot of this is that Non-Negativity is superfluous once we have specified the representation of credences that we are using. But another is that Probabilism incorporates not only normative claims, such as Normalization I and II and Finite Additivity, but also a metaphysical claim, namely, that there is a lowest possible credence that you can have and a highest possible credence that you can have. Without that, we couldn't specify the representation of credences in such a way that we would want to sign up to Normalization I and II. Suppose that, for any credence you can have, there is a higher one than you could have. Then there is no credence that I would want to demand you have in a necessary truth--for any I demanded, it would be better for you to have one higher. So I either have to say that all credences in necessary falsehoods are rationally forbidden, or all are rationally permitted, or I pick some threshold above which any credence is rationally permitted. And the same goes, mutatis mutandis, for credences in necessary falsehoods. I'm not sure what the norm of credences would be if our credences were unbounded in one or other or both directions. But it certainly wouldn't be Probabilism.<br /><br />So Non-Negativity is not a normative claim, but rather a trivial consequence of a metaphysical claim together with a conventional choice of representation. The metaphysical claim is that there is a minimal and a maximal credence; the representation choice is that 0 will represent the minimal credence and 1 will represent the maximal credence.<br /><br />Next, suppose we make a different conventional choice. Suppose we pick real numbers $a$ and $b$, and we say that $a$ represents minimal credence and $b$ represents maximal credence. Then clearly Normalization I becomes $c(\bot) = a$ and Normalization II becomes $c(\top) = b$. But what of Finite Additivity? This looks problematic. After all, if $a = 10$ and $b = 30$, and $c(A) = 20 = c(\overline{A})$, then Finite Addivitity demands that $c(\top) = c(A \vee \overline{A}) = c(A) + c(\overline{A}) = 40$, which is greater than the maximal credence. So Finite Additivity makes an impossible demand on an agent who seems to have perfectly rational credences in $A$ and $\overline{A}$, given the representation.<br /><br />The reason is that Finite Additivity, formulated as we formulated it above, is peculiar to very specific representations of credences, such as the standard one on which 0 stands for minimal credence and 1 stands for maximal credence. The correct formulation of Finite Additivity in general says: $c(A \vee B) = c(A) + c(B) - c(A\ \&\ B)$, for any propositions $A$, $B$ in $\mathcal{F}$. Thus, in the case we just gave above, if $c(A\ \&\ \overline{A}) = 10$, in keeping with the relevant version of Normalization I, we have $c(A \vee \overline{A}) = 20 + 20 - 10 = 30$, as required. So we see that it's wrong to say that Probabilism says that your credence in the disjunction of two mutually exclusive propositions should be the sum of your credences in the disjuncts--that's actually only true on some representation of your credences (namely, those for which 0 represents minimal credence).<br /><br />Bringing all of this together, I propose the following formulation of Probabilism:<br /><br /><b>Probabilism (revised formulation)</b> <br /><ul><li>(Bounded credences) There is a lowest possible credence you can have; and there is a highest possible credence you can have.</li><li>(Representation) We represent the lowest possible credence you have using $a$, and we represent the highest possible credence you can have using $b$.</li><li>(Normalization I) Your credence in a necessarily false proposition should be the lowest possible credence you can have. In symbols: $c(\bot) = a$.</li><li>(Normalization II) Your credence in a necessarily true proposition should be the highest possible credence you can have. In symbols: $c(\top) = b$.</li><li>(Finite Additivity) $c(A \vee B) = c(A) + c(B) - c(A\ \&\ B)$, for any propositions $A$, $B$ in $\mathcal{F}$.</li></ul>We call such a credence function a probability$_{a, b}$ function. How can we be sure this is right? Here are some considerations in its favour:<br /><br /><b>Switching representations </b><br />(i)<b> </b>Suppose $c(-)$ is a probability$_{a, b}$ function. Then $\frac{1}{b-a}c(-) - \frac{a}{b-a}$ is a probability function (or probability$_{0, 1}$ function).<br />(ii) Suppose $c(-)$ is a probability function and $a, b$ are real numbers. Then $c(-)(b-a) + a$ is a probability$_{a, b}$ function.<br /><br /><b>Dutch Book Argument</b><br />The standard Dutch Book Argument for Probabilism assumes that, if you have credence $p$ in proposition $X$, then you will pay £$pS$ for a bet that pays £$S$ if $X$ and £$0$ if $\overline{X}$. But this assumes that you have credences between 0 and 1, inclusive. What is the corresponding assumption if you represent credences in a different scale? Shorn of its conventional choice of representation, the assumption is: (a) you will pay £$0$ for a bet on $X$ if you have minimal credence in $X$; (b) you will pay £$S$ for a bet on $X$ if you have maximal credence in $X$; (c) the price you will pay for a bet on $X$ increases linearly with your credence in $X$. Translated into a framework in which we measure credence on a scale from $a$ to $b$, the assumption is then: you will pay £$\frac{p-a}{b-a}S$ for a bet that pays £$S$ if $X$ and £$0$ if $\overline{X}$. And, with this assumption, we can find Dutch Books against any credence function that isn't a probability$_{a, b}$ function.<br /><br /><b>Accuracy Dominance Argument</b><br />The standard Accuracy Dominance Argument for Probabilism assumes that, for each world, the ideal or vindicated credence function at that world assigns 0 to all falsehoods and 1 to all truths. Of course, if we represent minimal credence by $a$ and maximal credence by $b$, then we'll want to change that assumption. We'll want to say instead that the ideal or vindicated credence function at a world assigns $a$ to falsehoods and $b$ to truths. Once we say that, for any credence function that isn't a probability$_{a, b}$ function, there is another credence function that is closer to the ideal credence function at all worlds.<br /><br />So, the usual arguments for having a credence function that is a probability function when you represent your credences on a scale from 0 to 1 can be repurposed to argue that you should have a credence function that is a probability$_{a, b}$ function when you represent your credences on a scale from $a$ to $b$. And that gives us good reason to think that the second formulation of Probabilism above is correct.<br /><br /><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-52914705473919112162018-07-16T11:02:00.000+01:002018-07-16T11:02:49.107+01:00Yet another assistant professorship in formal philosophy @ University of Gdansk<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: left;"><a href="http://entiaetnomina.blogspot.com/2016/12/assistant-professorship-in-mathematical.html" style="text-align: justify;" target="_blank">Some time ago</a><span style="text-align: justify;"> the Chair of Logic, Philosophy of Science and Epistemology had an opening in formal philosophy that since then has been filled. Now, another position (leading to a permanent position upon second renewal) is available (so, there'll be three tenure-track faculty members working on formal philosophy). <a href="http://entiaetnomina.blogspot.com/2018/07/yet-another-assistant-professorship-in.html">Details.</a></span></div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-59484165495972496612018-07-16T08:49:00.003+01:002018-07-16T08:49:58.175+01:00Lecturer Position in Logic and Philosophy of Language (MCMP) <br /><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">The Ludwig-Maximilians-University Munich is one of the largest and most prestigious universities in Germany.</span><span lang="EN-US" style="mso-ansi-language: EN-US;"></span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">Ludwig-Maximilians-University Munich is seeking applications for one</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: center;"><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="mso-ansi-language: EN-US;">Lecturer Position (equivalent to Assistant Professorship) </span></b></div><div class="MsoPlainText" style="text-align: center;"><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="mso-ansi-language: EN-US;">in Logic and Philosophy of Language</span></b></div><div class="MsoPlainText" style="text-align: center;"><span lang="EN-US" style="mso-ansi-language: EN-US;">(for three years, with the possibility of extension)</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">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 December 1, 2018, is for three years with the possibility of extension for another three years.</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">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.</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">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).</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">Your workplace is centrally located in Munich and is very easy to reach by public transport. We offer you an interesting and responsible job with good training and development opportunities.</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">The employment takes place within the TV-L scheme.</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">The position is initially limited to November 30, 2021.</span></div><div class="MsoNormal" style="line-height: normal; text-align: left;"><span lang="EN-US" style="font-family: "Calibri",sans-serif; letter-spacing: 0pt; mso-ansi-language: EN-US;">Furthermore, given equal qualification, severely physically challenged applicants will be preferred</span><span lang="EN" style="mso-ansi-language: EN;">.</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">There is the possibility of part-time employment.</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">The application of women is strongly welcome.</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">Applications (including CV, certificates, list of publications, list of courses taught, a writing sample and a description of planned research projects (1000-1500 words)) should be sent either by email (ideally all requested documents in just one PDF document) or by mail to</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">Ludwig-Maximilians-Universität München</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">Faculty of Philosophy, Philosophy of Science and Study of Religion</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">Chair of Logic and Philosophy of Language / MCMP</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="DE">Geschwister-Scholl-Platz 1</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="DE">80539 München</span></div><div class="MsoPlainText" style="text-align: left;"><span lang="DE">e-Mail: </span><span class="MsoHyperlink"><span lang="DE" style="font-family: "Calibri",sans-serif; mso-bidi-font-family: "Times New Roman";"><a href="mailto:office.leitgeb@lrz.uni-muenchen.de">office.leitgeb@lrz.uni-muenchen.de</a></span></span><span lang="DE"></span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">by <b style="mso-bidi-font-weight: normal;">September 1, 2018</b>. If possible, we very much prefer applications by email.</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">In addition, we ask for two letters if reference, which must be sent by the reviewers directly to the above address (e-mail preferred).</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN" style="mso-ansi-language: EN;">For further questions you can contact by e-mail <span class="MsoHyperlink"><span style="font-family: "Calibri",sans-serif; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi;"><a href="mailto:office.leitgeb@lrz.uni-muenchen.de">office.leitgeb@lrz.uni-muenchen.de</a></span></span>.</span><span lang="EN-US" style="mso-ansi-language: EN-US;"></span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div class="MsoPlainText" style="text-align: left;"><span lang="EN-US" style="mso-ansi-language: EN-US;">More information about the MCMP can be found at </span><span class="MsoHyperlink"><span lang="EN-US" style="font-family: "Calibri",sans-serif; mso-ansi-language: EN-US; mso-bidi-font-family: "Times New Roman";"><a href="http://www.mcmp.philosophie.uni-muenchen.de/index.html">http://www.mcmp.philosophie.uni-muenchen.de/index.html</a></span></span><span lang="EN-US" style="mso-ansi-language: EN-US;">.</span></div><div class="MsoPlainText" style="text-align: left;"><br /></div><div style="text-align: left;"> <span lang="EN-US" style="font-family: "Calibri",sans-serif; font-size: 11.0pt; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "LMU CompatilFact"; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: DE; mso-hansi-theme-font: minor-latin;">The German description of the position is to be found at <span class="MsoHyperlink"><span style="font-family: "Calibri",sans-serif; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "LMU CompatilFact"; mso-hansi-theme-font: minor-latin;"><a href="https://www.uni-muenchen.de/aktuelles/stellenangebote/wissenschaft/20180704161330.html">https://www.uni-muenchen.de/aktuelles/stellenangebote/wissenschaft/20180704161330.html</a></span></span></span> </div><div style="text-align: left;"><br /></div><style><!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; 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mso-style-unhide:no; font-family:"Times New Roman",serif; mso-bidi-font-family:"Times New Roman"; color:blue; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-noshow:yes; mso-style-priority:99; color:purple; mso-themecolor:followedhyperlink; text-decoration:underline; text-underline:single;} p.MsoPlainText, li.MsoPlainText, div.MsoPlainText {mso-style-priority:99; mso-style-link:"Testo normale Carattere"; margin:0cm; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; mso-bidi-font-size:10.5pt; font-family:"Calibri",sans-serif; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-ansi-language:DE; mso-fareast-language:EN-US;} span.TestonormaleCarattere {mso-style-name:"Testo normale Carattere"; mso-style-priority:99; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Testo normale"; mso-bidi-font-size:10.5pt; font-family:"Calibri",sans-serif; mso-ascii-font-family:Calibri; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:11.0pt; mso-ansi-font-size:11.0pt; mso-bidi-font-size:11.0pt; mso-ansi-language:DE; mso-fareast-language:DE;} @page WordSection1 {size:612.0pt 792.0pt; margin:70.85pt 2.0cm 2.0cm 2.0cm; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --></style>Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-67854095470407479962018-07-03T15:41:00.002+01:002018-07-03T15:41:33.894+01:00Entia et Nomina 2018 (August 28-29, Gdansk)<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: justify;">This is the seventh conference in the Entia et NominA series (previous editions took place in Poland, Belgium and India), which features workshops for researchers in formally and analytically oriented philosophy, in particular in <b>epistemology, logic, and philosophy of science</b>. The distinctive format of the workshop requires participants to distribute extended abstracts or full papers a couple of weeks before the workshop and to prepare extended comments on another participant's paper.</div><div><br /></div><div><b>Invited speakers</b></div><div>- Zalan Gyenis (Jagiellonian University, Poland)</div><div>- Masashi Kasaki (Nagoya University, Japan)</div><div>- Martin Smith (University of Edinburgh, Scotland)</div><div><br /></div><div><br /></div><div><b>Dates: </b></div><div>- Submission deadline: July 20</div><div>- Decisions: August 1</div><div>- Workshop: August 28-29</div><div><br /></div><div>For more details on the workshop and submission, consult the pdf file with full CFP:</div><div><br /></div><div><a href="https://drive.google.com/file/d/1XMYKaVEDyidZ6W901gDrVVN3_vIk4k_6/view?usp=sharing" target="_blank">Entia et Nomina FULL CFP</a></div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-38613388398177303672018-02-12T20:31:00.000+00:002018-07-26T08:20:15.610+01:00An almost-Dutch Book argument for the Principal PrinciplePeople often talk about the synchronic <a href="https://plato.stanford.edu/entries/dutch-book/#BasiDutcBookArguForProb" target="_blank">Dutch Book argument for Probabilism</a> and the <a href="https://plato.stanford.edu/entries/dutch-book/#DiacDutcBookArgu" target="_blank">diachronic Dutch Strategy argument for Conditionalization</a>. But the synchronic Dutch Book argument for the Principal Principle is mentioned less. That's perhaps because, in one sense, there couldn't possibly be such an argument. As the Converse Dutch Book Theorem shows, providing you satisfy Probabilism, there can be no Dutch Book made against you -- that is, there is no sets of bets, each of which you will consider fair or favourable on its own, but which, when taken together, lead to a sure loss for you. So you can violate the Principal Principle without being vulnerable to a sure loss, providing your satisfy Probabilism. However, there is a related argument for the Principal Principle. And conversations with a couple of philosophers recently made me think it might be worth laying it out.<br /><br />Here is the result on which the argument is based:<br /><br />(I) Suppose your credences violate the Principal Principle but satisfy Probabilism. Then there is a book of bets and a price such that: (i) you consider that price favourable for that book -- that is, your subjective expectation of the total net gain is positive; (ii) every possible objective chance function considers that price unfavourable -- that is, the objective expectation of the total net gain is guaranteed to be negative.<br /><br />(II) Suppose your credences satisfy both the Principal Principle and Probabilism. Then there is no book of bets and a price such that: (i) you consider that price favourable for that book; (ii) every possible objective chance function considers that price unfavourable.<br /><br />Put another way:<br /><br />(I') Suppose your credences violate the Principal Principle. There are two actions $a$ and $b$ such that: you prefer $b$ to $a$, but every possible objective chance function prefers $a$ to $b$.<br /><br />(II') Suppose your credences satisfy the Principal Principle. For any two actions $a$ and $b$: if every possible objective chance function prefers $a$ to $b$, then you prefer $a$ to $b$.<br /><br />To move from (I) and (II) to (I') and (II'), let $a$ be the action of accepting the bets in $B$ and let $b$ be the action of rejecting them. <br /><br />The proof splits into two parts:<br /><br />(1) First, we note that a credence function $c$ satisfies the Principal Principle iff $c$ is in the closed convex hull of the set of possible chance functions.<br /><br />(2) Second, we prove that:<br /><br />(2I) If a probability function $c$ lies outside the closed convex hull of a set of probability functions $\mathcal{X}$, then there is a book of bets and a price such the expected total net gain from that book at that price by the lights of $c$ is positive, while the expected total net gain from that book at that price by the lights of each $p$ in $\mathcal{X}$ is negative.<br /><br />(2II) If a probability function $c$ lies inside the closed convex hull of a set of probability functions $\mathcal{X}$, then there is no book of bets and a price such the expected total net gain from that book at that price by the lights of $c$ is positive, while the expected total net gain from that book at that price by the lights of each $p$ in $\mathcal{X}$ is negative.<br /><br />Here's the proof of (2), which I lift from my <a href="https://drive.google.com/file/d/11hxCUJAKLk7_6_WARz56z6ITm9lX4U5y/view" target="_blank">recent justification of linear pooling</a> -- the same technique is applicable since the Principal Principle essentially says that you should set your credences by applying linear pooling to the possible objective chances.<br /><br />First:<br /><ul><li>Let $\Omega$ be the set of possible worlds</li><li>Let $\mathcal{F} = \{X_1, \ldots, X_n\}$ be the set of propositions over which our probability functions are defined. So each $X_i$ is a subset of $\Omega$.</li></ul>Now:<br /><ul><li>We represent a probability function $p$ defined on $\mathcal{F}$ as a vector in $\mathbb{R}^n$, namely, $p = \langle p(X_1), \ldots, p(X_n)\rangle$.</li><li>Given a proposition $X$ in $\mathcal{F}$ and a stake $S$ in $\mathbb{R}$, we define the bet $B_{X, S}$ as follows: $$B_{X, S}(\omega) = \left \{ \begin{array}{ll}<br />S & \mbox{if } \omega \in X \\<br />0 & \mbox{if } \omega \not \in X<br />\end{array}<br />\right.$$ So $B_{X, S}$ pays out $S$ if $X$ is true and $0$ if $X$ is false.</li><li>We represent the book of bets $\sum^n_{i=1} B_{X_i, S_i}$ as a vector in $\mathbb{R}^n$, namely, $S = \langle S_1, \ldots, S_n\rangle$. </li></ul><br /><b>Lemma 1</b><br />If $p$ is a probability function on $\mathcal{F}$, the expected payoff of the book of bets $\sum^n_{i=1} B_{X_i, S_i}$ by the lights of $p$ is $$S \cdot p = \sum^n_{i=1} p(X_i)S_i$$<br /><b>Lemma 2</b><br />Suppose $c$ is a probability function on $\mathcal{F}$, $\mathcal{X}$ is a set of probability functions on $\mathcal{F}$, and $\mathcal{X}^+$ is the closed convex hull of $\mathcal{X}$. Then, if $c \not \in \mathcal{X}^+$, then there is a vector $S$ and $\varepsilon > 0$ such that, for all $p$ in $\mathcal{X}$, $$S \cdot p < S \cdot c - \varepsilon$$<br /><i>Proof of Lemma</i> <i>2</i>. Suppose $c \not \in \mathcal{X}^+$. Then let $c^*$ be the closest point in $\mathcal{X}^+$ to $c$. Then let $S = c - c^*$. Then, for any $p$ in $\mathcal{X}$, the angle $\theta$ between $S$ and $p - c$ is obtuse and thus $\mathrm{cos}\, \theta < 0$. So, since $S \cdot (p - c) = ||S||\, ||x - p|| \mathrm{cos}\, \theta$ and $||S||, ||p - c|| > 0$, we have $S \cdot (p - c) < 0$. And hence $S \cdot p < S \cdot c$. What's more, since $\mathcal{X}^+$ is closed, $p$ is not a limit point of $\mathcal{X}^+$, and thus there is $\delta > 0$ such that $||p - c|| > \delta$ for all $p$ in $\mathcal{X}$. Thus, there is $\varepsilon > 0$ such that $S \cdot p < S \cdot c - \varepsilon$, for all $p$ in $\mathcal{X}$.<br /><br />We now derive (2I) and (2II) from Lemmas 1 and 2:<br /><br />Let $\mathcal{X}$ be the set of possible objective chance functions. If $c$ violates the Principal Principle, then $c$ is not in $\mathcal{X}^+$. Thus, by Lemma 2, there is a book of bets $\sum^n_{i=1} B_{X_i, S_i}$ and $\varepsilon > 0$ such that, for any objective chance function $p$ in $\mathcal{X}$, $S \cdot p < S \cdot c - \varepsilon$. By Lemma 1, $S \cdot p$ is the expected payout of the book of bets by the lights of $p$, while $S \cdot c$ is the expected payout of the book of bets by the lights of $c$. Now, suppose we were to offer an agent with credence function $c$ the book of bets $\sum^n_{i=1} B_{X_i, S_i}$ for the price of $S \cdot c - \frac{\varepsilon}{2}$. Then this would have positive expected payoff by the lights of $c$, but negative expected payoff by the lights of each $p$ in $\mathcal{X}$. This gives (2I).<br /><br />(2II) then holds because, when $c$ is in the closed convex hull of $\mathcal{X}$, its expectation of a random variable is in the closed convex hull of the expectations of that random variable by the lights of the probability functions in $\mathcal{X}$. Thus, if the expectation of a random variable is negative by the lights of all the probability functions in $\mathcal{X}$, then its expectation by the lights of $c$ is not positive.<br /><br /><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-63562356791457858572018-01-01T20:39:00.000+00:002018-07-26T08:19:47.707+01:00A Dutch Book argument for linear poolingOften, we wish to aggregate the probabilistic opinions of different agents. They might be experts on the effects of housing policy on people sleeping rough, for instance, and we might wish to produce from their different probabilistic opinions an aggregate opinion that we can use to guide policymaking. Methods for undertaking such aggregation are called <i>pooling operators</i>. They take as their input a sequence of probability functions $c_1, \ldots, c_n$, all defined on the same set of propositions, $\mathcal{F}$. And they give as their output a single probability function $c$, also defined on $\mathcal{F}$, which is the aggregate of $c_1, \ldots, c_n$. (If the experts have non-probabilistic credences and if they have credences defined on different sets of propositions or events, problems arise -- I've written about these <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/2017/09/aggregating-abstaining-experts.html" target="_blank">here</a>.) Perhaps the simplest are the <i>linear pooling operators</i>. Given a set of non-negative weights, $\alpha_1, \ldots, \alpha_n \leq 1$ that sum to 1, one for each probability function to be aggregated, the linear pool of $c_1, \ldots, c_n$ with these weights is: $c = \alpha_1 c_1 + \ldots + \alpha_n c_n$. So the probability that the aggregate assigns to a proposition (or event) is the weighted average of the probabilities that the individuals assign to that proposition (event) with the weights $\alpha_1, \ldots, \alpha_n$.<br /><br />Linear pooling has had a hard time recently. <a href="http://onlinelibrary.wiley.com/doi/10.1111/nous.12143/abstract" target="_blank">Elkin and Wheeler</a> reminded us that linear pooling almost never preserves unanimous judgments of independence; <a href="https://link.springer.com/article/10.1007/s11098-014-0350-8" target="_blank">Russell, et al.</a> reminded us that it almost never commutes with Bayesian conditionalization; and <a href="http://eprints.lse.ac.uk/80762/1/Bradley_Learning%20from%20others_2017.pdf" target="_blank">Bradley</a> showed that aggregating a group of experts using linear pooling almost never gives the same result as you would obtain from updating your own probabilities in the usual Bayesian way when you learn the probabilities of those experts. I've tried to defend linear pooling against the first two attacks <a href="https://drive.google.com/file/d/0B-Gzj6gcSXKrWHNLZzF6TERraWc/view" target="_blank">here</a>. In that paper, I also offer a positive argument in favour of that aggregation method: I argue that, if your aggregate is not a result of linear pooling, there will be an alternative aggregate that each experts expects to be more accurate than yours; if your aggregate is a result of linear pooling, this can't happen. Thus, my argument is a non-pragmatic, accuracy-based argument, in the same vein as Jim Joyce's non-pragmatic vindication of probabilism. In this post, I offer an alternative, pragmatic, Dutch book-style defence, in the same vein as the standard Ramsey-de Finetti argument for probabilism.<br /><br />My argument is based on the following fact: <b>if your aggregate probability function is not a result of linear pooling, there will be a series of bets that the aggregate will consider fair but which each expert will expect to lose money (or utility); if your aggregate is a result of linear pooling, this can't happen.</b> Since one of the things we might wish to use an aggregate to do is to help us make communal decisions, a putative aggregate cannot be considered acceptable if it will lead us to make a binary choice one way when every expert agrees that it should be made the other way. Thus, we should aggregate credences using a linear pooling operator.<br /><br />We now prove the mathematical fact behind the argument, namely, that if $c$ is not a linear pool of $c_1, \ldots, c_n$, then there is a bet that $c$ will consider fair, and yet each $c_i$ will expect it to lose money; the converse is straightforward.<br /><br />Suppose $\mathcal{F} = \{X_1, \ldots, X_m\}$. Then:<br /><ul><li>We can represent a probability function $c$ on $\mathcal{F}$ as a vector in $\mathbb{R}^m$, namely, $c = \langle c(X_1), \ldots, c(X_m)\rangle$.</li><li>We can also represent a book of bets on the propositions in $\mathcal{F}$ by a vector in $\mathbb{R}^m$, namely, $S = \langle S_1, \ldots, S_m\rangle$, where $S_i$ is the stake of the bet on $X_i$, so that the bet on $X_i$ pays out $S_i$ dollars (or utiles) if $X_i$ is true and $0$ dollars (or utiles) if $X_i$ is false.</li><li>An agent with probability function $c$ will be prepared to pay $c(X_i)S_i$ for a bet on $X_i$ with stake $S_i$, and thus will be prepared to pay $S \cdot c = c(X_1)S_1 + \ldots + c(X_m)S_m$ dollars (or utiles) for the book of bets with stakes $S = \langle S_1, \ldots, S_m\rangle$. (As is usual in Dutch book-style arguments, we assume that the agent is risk neutral.)</li><li>This is because $S \cdot c$ is the expected pay out of the book of bets with stakes $S$ by the lights of probability function $c$.</li></ul>Now, suppose $c$ is not a linear pool of $c_1, \ldots, c_n$. So $c$ lies outside the convex hull of $\{c_1, \ldots, c_n\}$. Let $c^*$ be the closest point to $c$ inside that convex hull. And let $S = c - c^*$. Then the angle $\theta$ between $S$ and $c_i - c$ is obtuse and thus $\mathrm{cos}\, \theta < 0$ (see diagram below). So, since $S \cdot (c_i - c) = ||S||\, ||c_i - c|| \mathrm{cos}\, \theta$ and $||S||, ||c_i - c|| \geq 0$, we have $S \cdot (c_i - c) < 0$. And hence $S \cdot c_i < S \cdot c$. But recall:<br /><ul><li>$S \cdot c$ is the amount that the aggregate $c$ is prepared to pay for the book of bets with stakes $S$; and </li><li>$S \cdot c_i$ is the expert $i$'s expected pay out of the book of bets with stakes $S$.</li></ul>Thus, each expert will expect that book of bets to pay out less than $c$ will be willing to pay for it.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Z4J-OXKKzu8/WkqZZPqWmBI/AAAAAAAAApQ/wwuZLqQwtzIUzt17WzSiE5sycbnfaOlFwCLcBGAs/s1600/IMG_3856.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://1.bp.blogspot.com/-Z4J-OXKKzu8/WkqZZPqWmBI/AAAAAAAAApQ/wwuZLqQwtzIUzt17WzSiE5sycbnfaOlFwCLcBGAs/s400/IMG_3856.JPG" width="400" /></a></div><br /><br /><br />Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com2tag:blogger.com,1999:blog-4987609114415205593.post-78601383681488027912017-10-10T18:47:00.000+01:002017-10-10T18:47:29.584+01:00Two Paradoxes of Belief (by Roy T Cook)This was posted originally at the <a href="https://www.blogger.com/The%20Liar%20paradox%20arises%20via%20considering%20the%20Liar%20sentence:%20%20L:%20L%20is%20not%20true.%20%20and%20then%20reasoning%20in%20accordance%20with%20the:%20%20T-schema:%20%20%E2%80%9C%CE%A6%20is%20true%20if%20and%20only%20if%20what%20%CE%A6%20says%20is%20the%20case.%E2%80%9D%20%20Along%20similar%20lines,%20we%20obtain%20the%20Montague%20paradox%20(or%20the%20%E2%80%9Cparadox%20of%20the%20knower%E2%80%9C)%20by%20considering%20the%20following%20sentence:%20%20M:%20M%20is%20not%20knowable.%20%20and%20then%20reasoning%20in%20accordance%20with%20the%20following%20two%20claims:%20%20Factivity:%20%20%E2%80%9CIf%20%CE%A6%20is%20knowable%20then%20what%20%CE%A6%20says%20is%20the%20case.%E2%80%9D%20%20Necessitation:%20%20%E2%80%9CIf%20%CE%A6%20is%20a%20theorem%20(i.e.%20is%20provable),%20then%20%CE%A6%20is%20knowable.%E2%80%9D%20%20Put%20in%20very%20informal%20terms,%20these%20results%20show%20that%20our%20intuitive%20accounts%20of%20truth%20and%20of%20knowledge%20are%20inconsistent.%20Much%20work%20in%20logic%20has%20been%20carried%20out%20in%20attempting%20to%20formulate%20weaker%20accounts%20of%20truth%20and%20of%20knowledge%20that%20(i)%20are%20strong%20enough%20to%20allow%20these%20notions%20to%20do%20substantial%20work,%20and%20(ii)%20are%20not%20susceptible%20to%20these%20paradoxes%20(and%20related%20paradoxes,%20such%20as%20Curry%20and%20Yablo%20versions%20of%20both%20of%20the%20above).%20A%20bit%20less%20well%20known%20that%20certain%20strong%20but%20not%20altogether%20implausible%20accounts%20of%20idealized%20belief%20also%20lead%20to%20paradox.%20%20The%20puzzles%20involve%20an%20idealized%20notion%20of%20belief%20(perhaps%20better%20paraphrased%20at%20%E2%80%9Crational%20commitment%E2%80%9D%20or%20%E2%80%9Cjustifiable%20belief%E2%80%9D),%20where%20one%20believes%20something%20in%20this%20sense%20if%20and%20only%20if%20(i)%20one%20explicitly%20believes%20it,%20or%20(ii)%20one%20is%20somehow%20committed%20to%20the%20claim%20even%20if%20one%20doesn%E2%80%99t%20actively%20believe%20it.%20Hence,%20on%20this%20understanding%20belief%20is%20closed%20under%20logical%20consequence%20%E2%80%93%20one%20believes%20all%20of%20the%20logical%20consequences%20of%20one%E2%80%99s%20beliefs.%20In%20particular,%20the%20following%20holds:%20%20B-Closure:%20%20%E2%80%9CIf%20you%20believe%20that,%20if%20%CE%A6%20then%20%CE%A8,%20and%20you%20believe%20%CE%A6,%20then%20you%20believe%20%CE%A8.%E2%80%9D%20%20Now,%20for%20such%20an%20idealized%20account%20of%20belief,%20the%20rule%20of%20B-Necessitation:%20%20B-Necessitation:%20%20%E2%80%9CIf%20%CE%A6%20is%20a%20theorem%20(i.e.%20is%20provable),%20then%20%CE%A6%20is%20believed.%E2%80%9D%20%20is%20extremely%20plausible%20%E2%80%93%20after%20all,%20presumably%20anything%20that%20can%20be%20proved%20is%20something%20that%20follows%20from%20things%20we%20believe%20(since%20it%20follows%20from%20nothing%20more%20than%20our%20axioms%20for%20belief).%20In%20addition,%20we%20will%20assume%20that%20our%20beliefs%20are%20consistent:%20%20B-Consistency:%20%20%E2%80%9CIf%20I%20believe%20%CE%A6,%20then%20I%20do%20not%20believe%20that%20%CE%A6%20is%20not%20the%20case.%E2%80%9D%20%20So%20far,%20so%20good.%20But%20neither%20the%20belief%20analogue%20of%20the%20T-schema:%20%20B-schema:%20%20%E2%80%9C%CE%A6%20is%20believed%20if%20and%20only%20if%20what%20%CE%A6%20says%20is%20the%20case.%E2%80%9D%20%20nor%20the%20belief%20analogue%20of%20Factivity:%20%20B-Factivity:%20%20%E2%80%9CIf%20you%20believe%20%CE%A6%20then%20what%20%CE%A6%20says%20is%20the%20case.%E2%80%9D%20%20is%20at%20all%20plausible.%20After%20all,%20just%20because%20we%20believe%20something%20(or%20even%20that%20the%20claim%20in%20question%20follows%20from%20what%20we%20believe,%20in%20some%20sense)%20doesn%E2%80%99t%20mean%20the%20belief%20has%20to%20be%20true!%20%20There%20are%20other,%20weaker,%20principles%20about%20belief,%20however,%20that%20are%20not%20intuitively%20implausible,%20but%20when%20combined%20with%20B-Closure,%20B-Necessitation,%20and%20B-Consistency%20lead%20to%20paradox.%20We%20will%20look%20at%20two%20principles%20%E2%80%93%20each%20of%20which%20captures%20a%20sense%20in%20which%20we%20cannot%20be%20wrong%20about%20what%20we%20think%20we%20don%E2%80%99t%20believe.%20%20The%20first%20such%20principle%20we%20will%20call%20the%20First%20Transparency%20Principle%20for%20Disbelief:%20%20TPDB1:%20%20%E2%80%9CIf%20you%20believe%20that%20you%20don%E2%80%99t%20believe%20%CE%A6%20then%20you%20don%E2%80%99t%20believe%20%CE%A6.%E2%80%9D%20%20In%20other%20words,%20although%20many%20of%20our%20beliefs%20can%20be%20wrong,%20according%20to%20TPDB1%20our%20beliefs%20about%20what%20we%20do%20not%20believe%20cannot%20be%20wrong.%20The%20second%20principle,%20which%20is%20a%20mirror%20image%20of%20the%20first,%20we%20will%20call%20the%20Second%20Transparency%20Principle%20for%20Disbelief:%20%20TPDB2:%20%20%E2%80%9CIf%20you%20don%E2%80%99t%20believe%20%CE%A6%20then%20you%20believe%20that%20you%20don%E2%80%99t%20believe%20%CE%A6.%E2%80%9D%20%20In%20other%20words,%20according%20to%20TPDB2%20we%20are%20aware%20of%20(i.e.%20have%20true%20beliefs%20about)%20all%20of%20the%20facts%20regarding%20what%20we%20don%E2%80%99t%20believe.%20%20Either%20of%20these%20principles,%20combined%20with%20B-Closure,%20B-Necessitation,%20and%20B-Consistency,%20lead%20to%20paradox.%20I%20will%20present%20the%20argument%20for%20TPBD1.%20The%20argument%20for%20TPDB2%20is%20similar,%20and%20left%20to%20the%20reader%20(although%20I%20will%20give%20an%20important%20hint%20below).%20%20Consider%20the%20sentence:%20%20S:%20It%20is%20not%20the%20case%20that%20I%20believe%20S.%20%20Now,%20by%20inspection%20we%20can%20understand%20this%20sentence,%20and%20thus%20conclude%20that:%20%20(1)%20What%20S%20says%20is%20the%20case%20if%20and%20only%20if%20I%20do%20not%20believe%20S.%20%20Further,%20(1)%20is%20something%20we%20can,%20via%20inspecting%20the%20original%20sentence,%20informally%20prove.%20(Or,%20if%20we%20were%20being%20more%20formal,%20and%20doing%20all%20of%20this%20in%20arithmetic%20enriched%20with%20a%20predicate%20%E2%80%9CB(x)%E2%80%9D%20for%20idealized%20belief,%20a%20formal%20version%20of%20the%20above%20would%20be%20a%20theorem%20due%20to%20G%C3%B6del%E2%80%99s%20diagonalization%20lemma.)%20So%20we%20can%20apply%20B-Necessitation%20to%20(1),%20obtaining:%20%20(2)%20I%20believe%20that:%20what%20S%20says%20is%20the%20case%20if%20and%20only%20if%20I%20do%20not%20believe%20S.%20%20Applying%20a%20version%20of%20B-Closure,%20this%20entails:%20%20(3)%20I%20believe%20S%20if%20and%20only%20if%20I%20believe%20that%20I%20do%20not%20believe%20S.%20%20Now,%20assume%20(for%20reductio%20ad%20absurdum)%20that:%20%20(4)%20I%20believe%20S.%20%20Then%20combining%20(3)%20and%20(4)%20and%20some%20basic%20logic,%20we%20obtain:%20%20(5)%20I%20believe%20that%20I%20do%20not%20believe%20S.%20%20Applying%20TPDB1%20to%20(5),%20we%20get:%20%20(6)%20I%20do%20not%20believe%20S.%20%20But%20this%20contradicts%20(4).%20So%20lines%20(4)%20through%20(6)%20amount%20to%20a%20refutation%20of%20line%20(4),%20and%20hence%20a%20proof%20that:%20%20(7)%20I%20do%20not%20believe%20S.%20%20Now,%20(7)%20is%20clearly%20a%20theorem%20(we%20just%20proved%20it),%20so%20we%20can%20apply%20B-Necessitation,%20arriving%20at:%20%20(8)%20I%20believe%20that%20I%20do%20not%20believe%20S.%20%20Combining%20(8)%20and%20(3)%20leads%20us%20to:%20%20(9)%20I%20believe%20S.%20%20But%20this%20obviously%20contradicts%20(7),%20and%20we%20have%20our%20final%20contradiction.%20%20Note%20that%20this%20argument%20does%20not%20actually%20use%20B-Consistency%20(hint%20for%20the%20second%20argument%20involving%20TPDB2:%20you%20will%20need%20B-Consistency!)%20%20These%20paradoxes%20seem%20to%20show%20that,%20as%20a%20matter%20of%20logic,%20we%20cannot%20have%20perfectly%20reliable%20beliefs%20about%20what%20we%20don%E2%80%99t%20believe%20%E2%80%93%20in%20other%20words,%20in%20this%20idealized%20sense%20of%20belief,%20there%20are%20always%20things%20that%20we%20believe%20that%20we%20don%E2%80%99t%20believe,%20but%20in%20actuality%20we%20do%20believe%20(the%20failure%20of%20TPDB1),%20and%20things%20that%20we%20don%E2%80%99t%20believe,%20but%20don%E2%80%99t%20believe%20that%20we%20don%E2%80%99t%20believe%20(the%20failure%20of%20TPDB2).%20At%20least,%20the%20puzzles%20show%20this%20if%20we%20take%20them%20to%20force%20us%20to%20reject%20both%20TPDB1%20and%20TPDB2%20in%20the%20same%20way%20that%20many%20feel%20that%20the%20Liar%20paradox%20forces%20us%20to%20abandon%20the%20full%20T-Schema.%20%20Once%20we%E2%80%99ve%20considered%20transparency%20principles%20for%20disbelief,%20it%E2%80%99s%20natural%20to%20consider%20corresponding%20principles%20for%20belief.%20There%20are%20two.%20The%20first%20is%20the%20First%20Transparency%20Principle%20for%20Belief:%20%20TPB1:%20%20%E2%80%9CIf%20you%20believe%20that%20you%20believe%20%CE%A6%20then%20you%20believe%20%CE%A6.%E2%80%9D%20%20In%20other%20words,%20according%20to%20TPD1%20our%20beliefs%20about%20what%20we%20believe%20cannot%20be%20wrong.%20The%20second%20principle,%20again%20is%20a%20mirror%20image%20of%20the%20first,%20is%20the%20Second%20Transparency%20Principle%20for%20Belief:%20%20TPB2:%20%20%E2%80%9CIf%20you%20believe%20%CE%A6%20then%20you%20believe%20that%20you%20believe%20%CE%A6.%E2%80%9D%20%20In%20other%20words,%20according%20to%20TPB2%20we%20are%20aware%20of%20all%20of%20the%20facts%20regarding%20what%20we%20believe.%20%20Are%20either%20of%20these%20two%20principles,%20combined%20with%20B-Closure,%20B-Necessitation,%20and%20B-Consistency,%20paradoxical?%20If%20not,%20are%20there%20additional,%20plausible%20principles%20that%20would%20lead%20to%20paradoxes%20if%20added%20to%20these%20claims?%20I%E2%80%99ll%20leave%20it%20to%20the%20reader%20to%20explore%20these%20questions%20further.%20%20A%20historical%20note:%20Like%20so%20many%20other%20cool%20puzzles%20and%20paradoxes,%20versions%20of%20some%20of%20these%20puzzles%20first%20appeared%20in%20the%20work%20of%20medieval%20logician%20Jean%20Buridan.">OUPBlog</a>. This is a first in a series of cross-posted blogs by <a href="https://cla.umn.edu/about/directory/profile/cookx432" target="_blank">Roy T Cook</a> (Minnesota) from the OUPBlog series on <a href="https://blog.oup.com/category/series-columns/paradoxes-puzzles-roy-cook/" target="_blank">Paradox and Puzzles</a>.<br /><br />The Liar paradox arises via considering the Liar sentence:<br /><br />L: L is not true.<br /><br />and then reasoning in accordance with the:<br /><br />T-schema:<br /><br />“Φ is true if and only if what Φ says is the case.”<br /><br />Along similar lines, we obtain the Montague paradox (or the “paradox of the knower“) by considering the following sentence:<br /><br />M: M is not knowable.<br /><br />and then reasoning in accordance with the following two claims:<br /><br />Factivity:<br /><br />“If Φ is knowable then what Φ says is the case.”<br /><br />Necessitation:<br /><br />“If Φ is a theorem (i.e. is provable), then Φ is knowable.”<br /><br />Put in very informal terms, these results show that our intuitive accounts of truth and of knowledge are inconsistent. Much work in logic has been carried out in attempting to formulate weaker accounts of truth and of knowledge that (i) are strong enough to allow these notions to do substantial work, and (ii) are not susceptible to these paradoxes (and related paradoxes, such as Curry and Yablo versions of both of the above). A bit less well known that certain strong but not altogether implausible accounts of idealized belief also lead to paradox.<br /><br />The puzzles involve an idealized notion of belief (perhaps better paraphrased at “rational commitment” or “justifiable belief”), where one believes something in this sense if and only if (i) one explicitly believes it, or (ii) one is somehow committed to the claim even if one doesn’t actively believe it. Hence, on this understanding belief is closed under logical consequence – one believes all of the logical consequences of one’s beliefs. In particular, the following holds:<br /><br />B-Closure:<br /><br />“If you believe that, if Φ then Ψ, and you believe Φ, then you believe Ψ.”<br /><br />Now, for such an idealized account of belief, the rule of B-Necessitation:<br /><br />B-Necessitation:<br /><br />“If Φ is a theorem (i.e. is provable), then Φ is believed.”<br /><br />is extremely plausible – after all, presumably anything that can be proved is something that follows from things we believe (since it follows from nothing more than our axioms for belief). In addition, we will assume that our beliefs are consistent:<br /><br />B-Consistency:<br /><br />“If I believe Φ, then I do not believe that Φ is not the case.”<br /><br />So far, so good. But neither the belief analogue of the T-schema:<br /><br />B-schema:<br /><br />“Φ is believed if and only if what Φ says is the case.”<br /><br />nor the belief analogue of Factivity:<br /><br />B-Factivity:<br /><br />“If you believe Φ then what Φ says is the case.”<br /><br />is at all plausible. After all, just because we believe something (or even that the claim in question follows from what we believe, in some sense) doesn’t mean the belief has to be true!<br /><br />There are other, weaker, principles about belief, however, that are not intuitively implausible, but when combined with B-Closure, B-Necessitation, and B-Consistency lead to paradox. We will look at two principles – each of which captures a sense in which we cannot be wrong about what we think we don’t believe.<br /><br />The first such principle we will call the First Transparency Principle for Disbelief:<br /><br />TPDB1:<br /><br />“If you believe that you don’t believe Φ then you don’t believe Φ.”<br /><br />In other words, although many of our beliefs can be wrong, according to TPDB1 our beliefs about what we do not believe cannot be wrong. The second principle, which is a mirror image of the first, we will call the Second Transparency Principle for Disbelief:<br /><br />TPDB2:<br /><br />“If you don’t believe Φ then you believe that you don’t believe Φ.”<br /><br />In other words, according to TPDB2 we are aware of (i.e. have true beliefs about) all of the facts regarding what we don’t believe.<br /><br />Either of these principles, combined with B-Closure, B-Necessitation, and B-Consistency, lead to paradox. I will present the argument for TPBD1. The argument for TPDB2 is similar, and left to the reader (although I will give an important hint below).<br /><br />Consider the sentence:<br /><br />S: It is not the case that I believe S.<br /><br />Now, by inspection we can understand this sentence, and thus conclude that:<br /><br />(1) What S says is the case if and only if I do not believe S.<br /><br />Further, (1) is something we can, via inspecting the original sentence, informally prove. (Or, if we were being more formal, and doing all of this in arithmetic enriched with a predicate “B(x)” for idealized belief, a formal version of the above would be a theorem due to Gödel’s diagonalization lemma.) So we can apply B-Necessitation to (1), obtaining:<br /><br />(2) I believe that: what S says is the case if and only if I do not believe S.<br /><br />Applying a version of B-Closure, this entails:<br /><br />(3) I believe S if and only if I believe that I do not believe S.<br /><br />Now, assume (for reductio ad absurdum) that:<br /><br />(4) I believe S.<br /><br />Then combining (3) and (4) and some basic logic, we obtain:<br /><br />(5) I believe that I do not believe S.<br /><br />Applying TPDB1 to (5), we get:<br /><br />(6) I do not believe S.<br /><br />But this contradicts (4). So lines (4) through (6) amount to a refutation of line (4), and hence a proof that:<br /><br />(7) I do not believe S.<br /><br />Now, (7) is clearly a theorem (we just proved it), so we can apply B-Necessitation, arriving at:<br /><br />(8) I believe that I do not believe S.<br /><br />Combining (8) and (3) leads us to:<br /><br />(9) I believe S.<br /><br />But this obviously contradicts (7), and we have our final contradiction.<br /><br />Note that this argument does not actually use B-Consistency (hint for the second argument involving TPDB2: you will need B-Consistency!)<br /><br />These paradoxes seem to show that, as a matter of logic, we cannot have perfectly reliable beliefs about what we don’t believe – in other words, in this idealized sense of belief, there are always things that we believe that we don’t believe, but in actuality we do believe (the failure of TPDB1), and things that we don’t believe, but don’t believe that we don’t believe (the failure of TPDB2). At least, the puzzles show this if we take them to force us to reject both TPDB1 and TPDB2 in the same way that many feel that the Liar paradox forces us to abandon the full T-Schema.<br /><br />Once we’ve considered transparency principles for disbelief, it’s natural to consider corresponding principles for belief. There are two. The first is the First Transparency Principle for Belief:<br /><br />TPB1:<br /><br />“If you believe that you believe Φ then you believe Φ.”<br /><br />In other words, according to TPD1 our beliefs about what we believe cannot be wrong. The second principle, again is a mirror image of the first, is the Second Transparency Principle for Belief:<br /><br />TPB2:<br /><br />“If you believe Φ then you believe that you believe Φ.”<br /><br />In other words, according to TPB2 we are aware of all of the facts regarding what we believe.<br /><br />Are either of these two principles, combined with B-Closure, B-Necessitation, and B-Consistency, paradoxical? If not, are there additional, plausible principles that would lead to paradoxes if added to these claims? I’ll leave it to the reader to explore these questions further.<br /><br />A historical note: Like so many other cool puzzles and paradoxes, versions of some of these puzzles first appeared in the work of medieval logician Jean Buridan.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com2tag:blogger.com,1999:blog-4987609114415205593.post-8893471410512231962017-09-10T09:49:00.001+01:002017-09-10T09:49:19.900+01:00Aggregating abstaining expertsIn a series of posts a few months ago (<a href="http://m-phi.blogspot.co.uk/2017/03/a-dilemma-for-judgment-aggregation.html" target="_blank">here</a>, <a href="http://m-phi.blogspot.co.uk/2017/03/a-little-more-on-aggregating-incoherent.html" target="_blank">here</a>, and <a href="http://m-phi.blogspot.co.uk/2017/03/aggregating-incoherent-credences-case.html" target="_blank">here</a>), I explored a particular method by which we might aggregate expert credences when those credences are incoherent. The result was this <a href="https://drive.google.com/file/d/0B-Gzj6gcSXKrSTRZRGNxOUdIR3M/view?usp=sharing" target="_blank">paper</a>, which is now forthcoming in <i>Synthese</i>. The method in question was called <i>the coherent approximation principle</i> (CAP), and it was introduced by Daniel Osherson and Moshe Vardi in <a href="https://www.cs.rice.edu/~vardi/papers/geb06.pdf" target="_blank">this</a> 2006 paper. CAP is based on what we might call <i>the principle of minimal mutilation</i>. We begin with a collection of credence functions, $c_1$, ..., $c_n$, one for each expert, and some of which might be incoherent. What we want at the end is a single coherent credence function $c$ that is the aggregate of $c_1$, ..., $c_n$. The principle of minimal mutilation says that $c$ should be as close as possible to the $c_i$s -- when aggregating a collection of credence functions, you should change them as little as possible to obtain your aggregate.<br /><br />We can spell this out more precisely by introducing a <i>divergence</i> $\mathfrak{D}$. We might think of this as a measure of how far one credence function lies from another. Thus, $\mathfrak{D}(c, c')$ measures the distance from $c$ to $c'$. We call these measures <i>divergences</i> rather than <i>distances</i> or <i>metrics</i>, since they do not have the usual features that mathematicians assume of a metric: we assume $\mathfrak{D}(c, c') \geq 0$, for any $c, c'$, and $\mathfrak{D}(c, c') = 0$ iff $c = c'$, but we do not assume that $\mathfrak{D}$ is symmetric nor that it satisfies the triangle inequality. In particular, we assume that $\mathfrak{D}$ is an <i>additive Bregman divergence</i>. The standard example of an additive Bregman divergence is <i>squared Euclidean distance</i>: if $c$, $c'$ are both defined on the set of propositions $F$, then<br />$$<br />\mathrm{SED}(c, c') = \sum_{X \in F} |c(X) - c'(X)|^2<br />$$In fact, $\mathrm{SED}$ is symmetric, but it does not satisfy the triangle inequality. The details of this family of divergences needn't detain us here (but see here and here for more). Indeed, we will simply use $\mathrm{SED}$ throughout. But a more general treatment would look at other additive Bregman divergences, and I hope to do this soon.<br /><br />Now, suppose $c_1$, ..., $c_n$ is a set of expert credence functions. And suppose $c_i$ is defined on the set of propositions $F_i$. And suppose that $\mathfrak{D}$ is an additive Bregman divergence -- you might take it to be $\mathrm{SED}$. Then how do we define the aggregate $c$ that is obtained from $c_1$, ..., $c_n$ by a minimal mutilation? We let $c$ be the coherent credence function such that the sum of the distances from $c$ to the $c_i$s is minimal. That is,<br />$$<br />\mathrm{CAP}_{\mathfrak{D}}(c_1, \ldots, c_n) = \mathrm{arg\ min}_{c \in P_{F_i}} \sum^n_{i=1} \mathfrak{D}(c, c_i)<br />$$<br />where $P_{F_i}$ is the set of coherent credence functions over $F_i$.<br /><br />As we see in my paper linked above, if each of the credence functions are defined over the same set of propositions -- that is, if $F_i = F_j$, for all $1 \leq i, j, \leq n$ -- then:<br /><ul><li>if $\mathfrak{D}$ is squared Euclidean distance, then this aggregate is the <i>straight linear pool</i> of the original credences; if $c$ is defined on the partition $X_1$, ..., $X_m$, then the straight linear pool of $c_1$, ..., $c_n$ is this:$$c(X_j) = \frac{1}{n}c_1(X_j) + ... + \frac{1}{n}c_n(X_j)$$</li><li>if $\mathfrak{D}$ is the generalized Kullback-Leibler divergence, then the aggregate is the <i>straight geometric pool</i> of the originals; if $c$ is defined on the partition $X_1$, ..., $X_m$, then the straight geometric pool of $c_1$, ..., $c_n$ is this: $$c(X_j) = \frac{1}{K}(c_1(X_j)^{\frac{1}{n}} \times ... \times c_1(X_j)^{\frac{1}{n}})$$where $K$ is a normalizing factor.</li></ul>(For more on these types of aggregation, see <a href="http://personal.lse.ac.uk/list/PDF-files/OpinionPoolingReview.pdf" target="_blank">here</a> and <a href="https://link.springer.com/article/10.1007/s11098-014-0350-8" target="_blank">here</a>).<br /><br />In this post, I'm interested in cases where our agents have credences in different sets of propositions. For instance, the first agent has credences concerning the rainfall in Bristol tomorrow and the rainfall in Bath, but the second has credences concerning the rainfall in Bristol and the rainfall in Birmingham.<br /><br />I want to begin by pointing to a shortcoming of CAP when it is applied to such cases. It fails to satisfy what we might think of as a basic desideratum of such procedures. To illustrate this desideratum, let's suppose that the three propositions $X_1$, $X_2$, and $X_3$ form a partition. And suppose that Amira has credences in $X_1$, $X_2$, and $X_3$, while Benito has credences only in $X_1$ and $X_2$. In particular:<br /><ul><li>Amira's credence function is: $c_A(X_1) = 0.3$, $c_A(X_2) = 0.6$, $c_A(X_3) = 0.1$.</li><li>Benito's credence function is: $c_B(X_1) = 0.2$, $c_B(X_2) = 0.6$.</li></ul>Now, notice that, while Amira's credence function is defined on the whole partition, Benito's is not. But, nonetheless, Benito's credences uniquely determine a coherent credence function on the whole partition:<br /><ul><li>Benito's extended credence function is: $c^*_B(X_1) = 0.2$, $c^*_B(X_2) = 0.6$, $c^*_B(X_3) = 0.2$.</li></ul>Thus, we might expect our aggregation procedure to give the same result whether we aggregate Amira's credence function with Benito's or with Benito's extended credence function. That is, we might expect the same result whether we aggregate $c_A$ with $c_B$ or with $c^*_B$. After all, $c^*_B$ is in some sense implicit in $c_B$. An agent with credence function $c_B$ is committed to the credences assigned by credence function $c^*_B$.<br /><br />However, CAP does not do this. As mentioned above, if you aggregate $c_A$ and $c^*_B$ using $\mathrm{SED}$, then the result is their linear pool: $\frac{1}{2}c_A + \frac{1}{2}c^*_B$. Thus, the aggregate credence in $X_1$ is $0.25$; in $X_2$ it is $0.6$; and in $X_3$ it is $0.15$. The result is different if you aggregate $c_A$ and $c_B$ using $SED$: the aggregate credence in $X_1$ is $0.2625$; in $X_2$ it is $0.6125$; in $X_3$ it is $0.125$.<br /><br />Now, it is natural to think that the problem arises here because Amira's credences are getting too much say in how far a potential aggregate lies from the agents, since she has credences in three propositions, while Benito only has credences in two. And, sure enough, $\mathrm{CAP}_{\mathrm{SED}}(c_A, c_B)$ lies closer to $c_A$ than to $c_B$ and closer to $c_A$ than the aggregate of $c_A$ and $c^*_B$ lies. And it is equally natural to try to solve this potential bias in favour of the agent with more credences by normalising. That is, we might define a new version of CAP:<br />$$<br />\mathrm{CAP}^+_D(c_1, \ldots, c_n) = \mathrm{arg\ min}_{c' \in P_{F_i}} \sum^n_{i=1} \frac{1}{|F_i|}D(c, c_i)<br />$$<br />However, this doesn't help. Using this definition, the aggregate of Amira's credence function $c_A$ and Benito's extended credence function $c^*_B$ remains the same; but the aggregate of Amira's credence function and Benito's original credence function changes -- the aggregate credence in $X_1$ is $0.25333$; in $X_2$, it is $0.61333$; in $X_3$, it is $0.1333$. Again, the two ways of aggregating disagree.<br /><br />So here is our desideratum in general:<br /><br /><b>Agreement with Coherent Commitments (ACC)</b> Suppose $c_1$, ..., $c_n$ are coherent credence functions, with $c_i$ defined on $F_i$, for each $1 \leq i \leq n$. And let $F = \bigcup^n_{i=1} F_i$. Now suppose that, for each $c_i$ defined on $F_i$, there is a unique coherent credence function $c^*_i$ defined on $F$ that extends $c_i$ -- that is, $c_i(X) = c^*_i(X)$ for all $X$ in $F_i$. Then the aggregate of $c_1$, ..., $c_n$ should be the same as the aggregate of $c^*_1$, ..., $c^*_n$.<br /><br />CAP does not satisfy ACC. Is there a natural aggregation rule that does? Here's a suggestion. Suppose you wish to aggregate a set of credence functions $c_1$, ..., $c_n$, where $c_i$ is defined on $F_i$, as above. Then we proceed as follows.<br /><ol><li>First, let $F = \bigcup^n_{i=1} F_i$.</li><li>Second, for each $1 \leq i \leq n$, let $$c^*_i = \{c : \mbox{$c$ is coherent & $c$ is defined on $F$ & $c(X) = c_i(X)$ for all $X$ in $F$}\}$$ That is, while $c_i$ represents a precise credal state defined on $F_i$, $c^*_i$ represents an imprecise credal state defined on $F$. It is the set of coherent credence functions on $F$ that extend $c_i$. That is, it is the set of coherent credence functions on $F$ that agree with $c_i$ on propositions in $F_i$. Thus, if, like Benito, your coherent credences on $F_i$ uniquely determine your coherent credences on $F$, then $c^*_i$ is just the singleton that contains that unique extension. But if your credences over $F_i$ do not uniquely determine your coherent credences over $F$, then $c^*_i$ will contain more coherent credence functions.</li><li>Finally, we take the aggregate of $c_1$, ..., $c_n$ to be the credence function $c$ that minimizes the total distance from $c$ to the $c^*_i$s. The problem is that there isn't a single natural definition of the distance from a point to a set of points, even when you have a definition of the distance between individual points. I adopt a very particular measure of such distances here; but it would be interesting to explore the alternative options in greater detail elsewhere. Suppose $c$ is a credence function and $C$ is a set of credence functions. Then $$D(c, C) = \frac{\mathrm{min}_{c' \in C}D(c, c') + \mathrm{max}_{c' \in C}D(c, c')}{2}$$ With this in hand, we can finally give our aggregation procedure:$$\mathrm{CAP}^*_D(c_1, \ldots, c_n) = \mathrm{arg\ min}_{c' \in P_F} \sum^n_{i=1} D(c, c^*_i)$$ </li></ol>The first thing to note about CAP$^*$ is that, unlike the original CAP, or CAP$^+$, it automatically satisfies ACC.<br /><br />Let's now see CAP$^*$ in action.<br /><ul><li>Since CAP$^*$ satisfies ACC, the aggregate for $c_A$ and $c_B$ is the same as the aggregate for $c_A$ and $c^*_B$, which is just their straight linear pool.</li><li>Next, suppose we wish to aggregate Amira with a third agent, Cleo, who has a credence only in $X_1$, which she assigns $0.5$ -- that is, $c_C(X_1) = 0.5$. Then $F = \{X_1, X_2, X_3\}$, and $$c^*_C = \{c : c(X_1) = 0.5, c(X_2) \geq 0.5, c(X_3) = 1 - c(X_1) - c(X_2)\}$$ So, $$\mathrm{CAP}^*_{\mathfrak{D}}(c_A, c_B) = \mathrm{arg\ min}_{c' \in P_F} \mathfrak{D}(c', c_A) + \mathfrak{D}(c', c^*_C)$$Working through the calculation for $\mathfrak{D} = \mathrm{SED}$, we obtain the following aggregate: $c(X_1) = 0.4$, $c(X_2) = 0.425$, $c(X_3) = 0.175$.</li><li>One interesting feature of CAP$^*$ is that, unlike CAP, we can apply it to individual agents. Thus, for instance, suppose we wish to take Cleo's single credence in $X_1$ and 'fill in' her credences in $X_2$ and $X_3$. Then we can use CAP$^*$ to do this. Her new credence function will be $$c'_C = \mathrm{CAP}^*_{\mathrm{SED}}(c_C) = \mathrm{arg\ min}_{c' \in P_F} D(c', c_C)$$ That is, $c'_C(X_1) = 0.5$, $c'_C(X_2) = 0.25$, $c'_C(X_3) = 0.25$. Rather unsurprisingly, $c'_C$ is the midpoint of the line formed by the imprecise probabilities $c^*_C$. Now, notice: the aggregate of Amira and Cleo given above is just the straight linear pool of Amira's credence function $c_A$ and Cleo's 'filled in' credence function $c'_C$. I would conjecture that this is generally true: filling in credences using CAP$^*_{\mathrm{SED}}$ and then aggregating using straight linear pooling always agrees with aggregating using CAP$^*_{\mathrm{SED}}$. And perhaps this generalises beyond SED.</li></ul>Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-63402577115091454602017-09-01T05:56:00.004+01:002017-09-01T05:56:52.697+01:00Two PhD positions in probability & law in Gdansk<div dir="ltr" style="text-align: left;" trbidi="on">More details <a href="http://entiaetnomina.blogspot.jp/2017/09/two-phd-positions-in-probability-law.html" target="_blank">here.</a></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-28586265974733149402017-07-31T16:49:00.000+01:002017-07-31T16:49:01.444+01:00Logic in the wild CFP (Ghent, 9-10 Nov 2017)<div dir="ltr" style="text-align: left;" trbidi="on"><div class="p1"><span class="s1"><b>CALL FOR PAPERS</b></span></div><div class="p1"><span class="s1"><b>workshop on </b></span></div><div class="p1"><span class="s1"><b>LOGIC IN THE WILD<span class="Apple-converted-space"> </span></b></span></div><div class="p1"><span class="s1"><b>Ghent University, 9 & 10 November 2017. </b></span></div><div class="p2"><span class="s1"></span><br /></div><div class="p3"><br /></div><div class="p3"><span class="s1"><b>The scope of this workshop</b></span></div><div class="p4" style="text-align: justify;"><span class="s1">Nowadays we are witnessing a ‘practical’, or cognitive turn in logic. The approach draws on enormous achievements of a legion of formal and mathematical logicians, but focuses on ‘the Wild’: actual human processes of reasoning and argumentation. Moreover, high standards of inquiry that we owe to formal logicians offer a new quality in research on reasoning and argumentation. In terms of John Corcoran’s distinction between logic as formal ontology and logic as formal epistemology, the aim of the practical turn is to make formal epistemology even more epistemically oriented. This is not to say that this ‘practically turned’ (or cognitively oriented) logic becomes just a part of psychology. This is to say that this logic acquires a new task of “systematically keeping track of changing representations of information”, as Johan van Benthem puts it, and that it contests the claim that the distinction between descriptive and normative accounts of reasoning is disjoint and exhaustive. From a different than purely psychological perspective logic becomes -- again -- interested in answering Dewey’s question about the Wild: how do we think? This is the new alluring face of psychologism, or cognitivism, in logic, as opposed to the old one, which Frege and Husserl fought against. This is the area of research to which our workshop is devoted.</span></div><div class="p3"><span class="s1">For this workshop we invite submissions on:</span></div><div class="p3"><span class="s1">- applications of logic to the analysis of actual human reasoning and argumentation processes.</span></div><div class="p3"><span class="s1">- tools and methods suited for such applications.</span></div><div class="p3"><span class="s1">- neural basis of logical reasoning.</span></div><div class="p3"><span class="s1">- educational issues of cognitively-oriented logic.</span></div><div class="p5"><span class="s1"></span><br /></div><div class="p3"><span class="s1"><b>Keynote speakers</b></span></div><div class="p6"><span class="s1">Keith Stenning (University of Edinburgh)</span></div><div class="p6"><span class="s1">Iris van Rooij (Radboud University Nijmegen)</span></div><div class="p3"><span class="s1">Christian Strasser (Ruhr University Bochum)</span></div><div class="p3"><br /></div><div class="p3"><span class="s1"><b>How to submit an abstract</b></span></div><div class="p3"><span class="s1">We welcome submissions on any topic that fits into the scope as described above. Send your abstract of 300 to 500 words to: <a href="mailto:lrr@ugent.be"><span class="s2">lrr@ugent.be</span></a> before <b>10 September 2017</b>.</span></div><div class="p3"><span class="s1">Notification of acceptance: 22 September 2017.</span></div><div class="p3"><br /></div><div class="p3"><span class="s1"><b>Website</b></span></div><div class="p3"><span class="s1">More information about the workshop (venue, registration, …) is available at</span></div><div class="p3"><span class="s2"><a href="http://www.lrr.ugent.be/logic-in-the-wild/">http://www.lrr.ugent.be/logic-in-the-wild/</a></span><span class="s1">. The programme will be available there in October.</span></div><div class="p3"><br /></div><div class="p3"><span class="s1"><b>Background</b></span></div><div class="p3"><span class="s1">This workshop is organized by the scientific research network <i>Logical and Methodological Analysis of Scientific Reasoning Processes</i> (LMASRP) which is sponsored by the Research Foundation Flanders (FWO).</span></div><div class="p3"><span class="s1">All information about the network can be found at <a href="http://www.lmasrp.ugent.be/"><span class="s2">http://www.lmasrp.ugent.be/</span></a></span></div><style type="text/css">p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px 'Times New Roman'; color: #212121; -webkit-text-stroke: #212121} p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; text-align: center; font: 12.0px 'Times New Roman'; color: #212121; -webkit-text-stroke: #212121; min-height: 15.0px} p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman'; color: #212121; -webkit-text-stroke: #212121} p.p4 {margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; font: 12.0px 'Times New Roman'; color: #212121; -webkit-text-stroke: #212121} p.p5 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman'; color: #212121; -webkit-text-stroke: #212121; min-height: 15.0px} p.p6 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman'; -webkit-text-stroke: #000000} span.s1 {font-kerning: none} span.s2 {text-decoration: underline ; font-kerning: none; color: #4787ff; -webkit-text-stroke: 0px #4787ff} </style> <br /><div class="p3"><span class="s1">An overview of the previous workshops of the network can be found at <a href="http://www.lrr.ugent.be/"><span class="s2">http://www.lrr.ugent.be/</span></a>.</span></div></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-65811373758714643312017-07-02T00:16:00.000+01:002017-07-02T00:16:22.600+01:00Three Postdoctoral Fellowships at the MCMP (LMU Munich)The Munich Center for Mathematical Philosophy (MCMP) seeks applications for <span class="dunkelrot">three 3-year postdoctoral fellowships</span> starting on <strong>October 1, 2017</strong>. (A later starting date is possible.) We are especially interested in candidates who work in the field of mathematical philosophy with a focus on philosophical logic (broadly construed, including philosophy and foundations of mathematics, semantics, formal philosophy of language, inductive logic and foundations of probability, and more).<br /><br /> Candidates who have not finished their PhD at the time of the application deadline have to provide evidence that they will have their PhD in hand at the time the fellowship starts. Applications (including a cover letter that addresses, amongst others, one's academic background, research interests and the proposed starting date, a CV, a list of publications, a sample of written work of no more than 5000 words, and a description of a planned research project of about 1000 words) should be sent by email (in one PDF document) to <a class="g-link-mail" href="mailto:office.leitgeb@lrz.uni-muenchen.de" title="Send email to: office.leitgeb@lrz.uni-muenchen.de">office.leitgeb@lrz.uni-muenchen.de</a> by <strong>August 15, 2017</strong>. Hard copy applications are not accepted. Additionally, two confidential letters of reference addressing the applicant's qualifications for academic research should be sent to the same email address from the referees directly.<br /><br /> The MCMP hosts a vibrant research community of faculty, postdoctoral fellows, doctoral fellows, master students, and visiting fellows. It organizes at least two weekly colloquia and a weekly internal work-in-progress seminar, as well as various other activities such as workshops, conferences, summer schools, and reading groups. The successful candidates will partake in the MCMP's academic activities and enjoy its administrative facilities and support. The official language at the MCMP is English and fluency in German is not mandatory.<br /><br /> We especially encourage female scholars to apply. The LMU in general, and the MCMP in particular, endeavor to raise the percentage of women among its academic personnel. Furthermore, given equal qualification, preference will be given to candidates with disabilities.<br /><br /> The fellowships are remunerated with 1.853 €/month (paid out without deductions for tax and social security). The MCMP is able to support fellows concerning expenses for professional traveling.<br /><br /> For further information, please contact <a href="http://www.mcmp.philosophie.uni-muenchen.de/people/faculty/hannes_leitgeb/index.html" title="Leitgeb, Hannes">Prof. Hannes Leitgeb</a> (<a class="g-link-mail" href="mailto:H.Leitgeb@lmu.de" title="Send email to: H.Leitgeb@lmu.de">H.Leitgeb@lmu.de</a>).<br /><br /><br /> Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-86370269246635121322017-07-02T00:13:00.001+01:002017-07-02T00:13:43.562+01:00Three Doctoral Fellowships at the MCMP (LMU Munich)The Munich Center for Mathematical Philosophy (MCMP) seeks applications for <span class="dunkelrot">three 3-year doctoral fellowships</span> starting on <strong>October 1, 2017</strong>. (A later starting date is possible.) We are especially interested in candidates who work in the field of mathematical philosophy with a focus on philosophical logic (broadly construed, including philosophy and foundations of mathematics, semantics, formal philosophy of language, inductive logic and foundations of probability, and more).<br /><br /> Candidates who have not finished their MA at the time of the application deadline have to provide evidence that they will have their MA in hand at the time the fellowship starts. Applications (including a cover letter that addresses, amongst others, one's academic background, research interests and the proposed starting date, a CV, a list of publications (if applicable), a sample of written work of no more than 3000 words, and a description of the planned PhD-project of about 2000 words) should be sent by email (in one PDF document) to <a class="g-link-mail" href="mailto:office.leitgeb@lrz.uni-muenchen.de" title="Send email to: office.leitgeb@lrz.uni-muenchen.de">office.leitgeb@lrz.uni-muenchen.de</a> by <strong>August 15, 2017</strong>. Hard copy applications are not accepted. Additionally, one confidential letter of reference addressing the applicant's qualifications for academic research should be sent to the same email address from the referees directly.<br /><br /> The MCMP hosts a vibrant research community of faculty, postdoctoral fellows, doctoral fellows, master students, and visiting fellows. It organizes at least two weekly colloquia and a weekly internal work-in-progress seminar, as well as various other activities such as workshops, conferences, summer schools, and reading groups. The successful candidates will partake in the MCMP's academic activities and enjoy its administrative facilities and support. The official language at the MCMP is English and fluency in German is not mandatory.<br /><br /> We especially encourage female scholars to apply. The LMU in general, and the MCMP in particular, endeavor to raise the percentage of women among its academic personnel. Furthermore, given equal qualification, preference will be given to candidates with disabilities.<br /><br /> The fellowships are remunerated with 1.468 €/month (paid out without deductions for tax and social security). The MCMP is able to support fellows concerning expenses for professional traveling.<br /><br /> For further information, please contact <a href="http://www.mcmp.philosophie.uni-muenchen.de/people/faculty/hannes_leitgeb/index.html" target="_blank" title="Leitgeb, Hannes">Prof. Hannes Leitgeb</a> (<a class="g-link-mail" href="mailto:H.Leitgeb@lmu.de" title="Send email to: H.Leitgeb@lmu.de">H.Leitgeb@lmu.de</a>).<br /><br /><br />Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com1tag:blogger.com,1999:blog-4987609114415205593.post-69091727360893763872017-05-16T12:06:00.000+01:002017-05-17T15:24:56.751+01:00The Wisdom of the Crowds: generalizing the Diversity Prediction TheoremI've just been reading <a href="http://aidanlyon.com/" target="_blank">Aidan Lyon</a>'s fascinating paper, <a href="http://aidanlyon.com/media/publications/WoCC.pdf" target="_blank">Collective Wisdom</a>. In it, he mentions a result known as the <i>Diversity Prediction Theorem</i>, which is sometimes taken to explain why crowds are wiser, on average, than the individuals who compose them. The theorem was originally proved by <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.8876" target="_blank">Anders Krogh and Jesper Vedelsby</a>, but it has entered the literature on social epistemology through the work of <a href="http://press.princeton.edu/titles/8757.html" target="_blank">Scott E. Page</a>. In this post, I'll generalize this result.<br /><br />The Diversity Prediction Theorem concerns a situation in which a number of different individuals estimate a particular quantity -- in the original example, it is the weight of an ox at a local fair. Take the crowd's estimate of the quantity to be the average of the individual estimates. Then the theorem shows that the distance from the crowd's estimate to the true value is less than the average distance from the individual estimates to the true value; and, moreover, the difference between the two is always given by the average distance from the individual estimates to the crowd's estimate (which you might think of as the variance of the individual estimates).<br /><br />Let's make this precise. Suppose you have a group of $n$ individuals. They each provide an estimate for a real-valued quantity. The $i^\mathrm{th}$ individual gives the prediction $q_i$. The true value of this quantity is $\tau$. And we measure the distance from one estimate of a quantity to another, or to the true value of that quantity, using squared error. Then:<br /><ul><li>The crowd's prediction of the quantity is $c = \frac{1}{n}\sum^n_{i=1} q_i$.</li><li>The crowd's distance from the true quantity is $\mathrm{SqE}(c) = (c-\tau)^2$.</li><li>$S_i$'s distance from the true quantity is $\mathrm{SqE}(q_i) = (q_i-\tau)^2$</li><li>The average individual distance from the true quantity is $\frac{1}{n} \sum^n_{i=1} \mathrm{SqE}(q_i) = \frac{1}{n} \sum^n_{i=1} (q_i - \tau)^2$.</li><li>The average individual distance from the crowd's estimate is $v = \frac{1}{n}\sum^n_{i=1} (q_i - c)^2$.</li></ul>Given this, we have:<br /><br /><b>Diversity Prediction Theorem</b> $$\mathrm{SqE}(c) = \frac{1}{n} \sum^n_{i=1} \mathrm{SqE}(q_i) - v$$ <br />The theorem is easy enough to prove. You essentially just follow the algebra. However, following through the proof, you might be forgiven for thinking that the result says more about some quirk of squared error as a measure of distance than about the wisdom of crowds. And of course squared error is just one way of measuring the distance from an estimate of a quantity to the true value of that quantity, or from one estimate of a quantity to another. There are other such distance measures. So the question arises: Does the Diversity Prediction Theorem hold if we replace squared error with one of these alternative measures of distance? In particular, it is natural to take any of the so-called Bregman divergences $\mathfrak{d}$ to be a legitimate measure of distance from one estimate to another. I won't say much about Bregman divergences here, except to give their formal definition. To learn about their properties, have a look <a href="http://mark.reid.name/blog/meet-the-bregman-divergences.html" target="_blank">here</a> and <a href="https://en.wikipedia.org/wiki/Bregman_divergence" target="_blank">here</a>. They were introduced by Bregman as a natural generalization of squared error.<br /><br /><b>Definition (Bregman divergence)</b> A function $\mathfrak{d} : [0, \infty) \times [0, \infty) \rightarrow [0, \infty]$ is a <i>Bregman divergence </i>if there is a continuously differentiable, strictly convex function $\varphi : [0, \infty) \rightarrow [0, \infty)$ such that $$\mathfrak{d}(x, y) = \varphi(x) - \varphi(y) - \varphi'(y)(x-y)$$<br />Squared error is itself one of the Bregman divergences. It is the one generated by $\varphi(x) = x^2$. But there are many others, each generated by a different function $\varphi$.<br /><br />Now, suppose we measure distance between estimates using a Bregman divergence $\mathfrak{d}$. Then:<br /><ul><li>The crowd's prediction of the quantity is $c = \frac{1}{n}\sum^n_{i=1} j_i$.</li><li>The crowd's distance from the true quantity is $\mathrm{E}(c) = \mathfrak{d}(c, \tau)$.</li><li>$S_i$'s distance from the true quantity is $\mathrm{E}(j_i) = \mathfrak{d}(q_i, \tau)$</li><li>The average individual distance from the true quantity is $\frac{1}{n} \sum^n_{i=1} \mathrm{E}(j_i) = \frac{1}{n} \sum^n_{i=1} \mathfrak{d}(q_i, \tau)$.</li><li>The average individual distance from the crowd's estimate is $v = \frac{1}{n}\sum^n_{i=1} \mathfrak{d}(q_i, c)$.</li></ul> Given this, we have:<br /><br /><b>Generalized Diversity Prediction Theorem</b> $$\mathrm{E}(c) = \frac{1}{n} \sum^n_{i=1} \mathrm{E}(q_i) - v$$<br /><i>Proof.</i><br />\begin{eqnarray*}<br />& & \frac{1}{n} \sum^n_{i=1} \mathrm{E}(q_i) - v \\<br />& = & \frac{1}{n} \sum^n_{i=1} [ \mathfrak{d}(q_i, \tau) - \mathfrak{d}(q_i, c)] \\ <br />& = & \frac{1}{n} \sum^n_{i=1} [\varphi(q_i) - \varphi(\tau) - \varphi'(\tau)(q_i - \tau)] - [\varphi(q_i) - \varphi(c) - \varphi'(\tau)(q_i - c)] \\<br />& = & \frac{1}{n} \sum^n_{i=1} [\varphi(q_i)- \varphi(\tau) - \varphi'(\tau)(q_i - \tau) - \varphi(q_i)+ \varphi(c) + \varphi'(\tau)(q_i - c)] \\<br />& = & - \varphi(\tau) - \varphi'(\tau)((\frac{1}{n} \sum^n_{i=1} q_i) - \tau) + \varphi(c) + \varphi'(\tau)((\frac{1}{n} \sum^n_{i=1} q_i) - c) \\<br />& = & - \varphi(\tau) - \varphi'(\tau)(c - \tau) + \varphi(c) + \varphi'(\tau)(c - c) \\<br />& = & \varphi(c) - \varphi(\tau) - \varphi'(\tau)(c - \tau) \\<br />& = & \mathfrak{d}(c, \tau) \\<br />& = & \mathrm{E}(c)<br />\end{eqnarray*}<br />as required.Richard Pettigrewhttp://www.blogger.com/profile/07828399117450825734noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-42608390793708542062017-05-11T07:44:00.001+01:002017-05-11T08:06:14.999+01:00Reasoning Club Conference 2017<br />The <b>Fifth Reasoning Club Conference</b> will take place at the <a href="https://www.llc.unito.it/" target="_blank">Center for Logic, Language, and Cognition</a> in Turin on May 18-19, 2017. <br /><br />The <a href="https://www.kent.ac.uk/secl/researchcentres/reasoning/club/index.html" 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 <i><a href="http://blogs.kent.ac.uk/thereasoner/about/" target="_blank">The Reasoner</a></i>. (Earlier editions of the meeting were held in <a href="http://www.vub.ac.be/CLWF/RC2012/" target="_blank">Brussels</a>, <a href="http://reasoningclubpisa.weebly.com/" target="_blank">Pisa</a>, <a href="https://reasoningclubkent.wordpress.com/" target="_blank">Kent</a>, and <a href="http://www.maths.manchester.ac.uk/news-and-events/events/fourth-reasoning-club-conf/" target="_blank">Manchester</a>.)<br /><br /><br /><br /><b>PROGRAM</b><br /><br /><br />THURSDAY, MAY 18<br /><br />Palazzo Badini<br />via Verdi 10, Torino<br />Sala Lauree di Psicologia (ground floor)<br /><br /><br />9:00 | welcome and coffee<br /><br />9:30 | greetings<br /> presentation of the new editorship of <i>The Reasoner</i><br /> (<b>Hykel HOSNI,</b> Milan)<br /><br /><br />Morning session – chair: Gustavo CEVOLANI (IMT Lucca)<br /><br /><br />10:00 | invited talk<br /><br /><b><a href="http://fitelson.org/" target="_blank">Branden FITELSON</a></b> (Northeastern University, Boston)<br /><br /><i>Two approaches to belief revision</i><br /><br />In this paper, we compare and contrast two methods for the qualitative revision of (viz., full) beliefs. The first (Bayesian) method is generated by a simplistic diachronic Lockean thesis requiring coherence with the agent's posterior credences after conditionalization. The second (Logical) method is the orthodox AGM approach to belief revision. Our primary aim will be to characterize the ways in which these two approaches can disagree with each other — especially in the special case where the agent's belief set is deductively cogent.<br /><br />(joint work with Ted Shear and Jonathan Weisberg)<br /><br /><br />11:00 | <b>Ted SHEAR</b> (Queensland) and <b>John QUIGGIN</b> (Queensland)<br /><i> </i><br /><i>A modal logic for reasonable belief</i><br /><br /><br />11:45 | <b>Nina POTH</b> (Edinburgh) and <b>Peter BRÖSSEL</b> (Bochum)<br /><br /><i>Bayesian inferences and conceptual spaces: Solving the complex-first paradox</i><br /><br /><br />12:30 | lunch break<br /><br /><br />Afternoon session I – chair: Peter BRÖSSEL (Bochum)<br /><br /><br />13:30 | invited talk<br /><br /><b><a href="https://www5.unitn.it/People/en/Web/Persona/PER0003393#INFO" target="_blank">Katya TENTORI</a></b> (University of Trento)<br /><br /><i>Judging forecasting accuracy </i><br /><i>How human intuitions can help improving formal models</i><br /><br />Most of the scoring rules that have been discussed and defended in the literature are not ordinally equivalent, with the consequence that, after the very same outcome has materialized, a forecast <i>X</i> can be evaluated as more accurate than <i>Y</i> according to one model but less accurate according to another. A question that naturally arises is therefore which of these models better captures people’s intuitive assessment of forecasting accuracy. To answer this question, we developed a new experimental paradigm for eliciting ordinal judgments of accuracy concerning pairs of forecasts for which various combinations of associations/dissociations between the Quadratic, Logarithmic, and Spherical scoring rules are obtained. We found that, overall, the Logarithmic model is the best predictor of people’s accuracy judgments, but also that there are cases in which these judgments — although they are normatively sound — systematically depart from what is expected by all the models. These results represent an empirical evaluation of the descriptive adequacy of the three most popular scoring rules and offer insights for the development of new formal models that might favour a more natural elicitation of truthful and informative beliefs from human forecasters.<br /><br />(joint work with Vincenzo Crupi and Andrea Passerini)<br /><br /><br />14:15 | <b>Catharine SAINT-CROIX</b> (Michigan)<br /><br /><i>Immodesty and evaluative uncertainty</i><br /><br /><br />15:15 | <b>Michael SCHIPPERS</b> (Oldenburg), <b>Jakob KOSCHOLKE</b> (Hamburg)<br /><br /><i>Against relative overlap measures of coherence</i><br /><br /><br />16:00 | coffee break<br /><br /><br />Afternoon session II – chair: Paolo MAFFEZIOLI (Torino)<br /><br /><br />16:30 | <b>Simon HEWITT</b> (Leeds)<br /><br /><i>Frege's theorem in plural logic</i><br /><br /><br />17:15 | <b>Lorenzo ROSSI</b> (Salzburg) and <b>Julien MURZI</b> (Salzburg)<br /><br /><i>Generalized Revenge</i><br /><br /><br /> <br />FRIDAY, MAY 19<br /><br />Campus Luigi Einaudi<br />Lungo Dora Siena 100/A<br />Sala Lauree Rossa<br />building D1 (ground floor)<br /><br /><br />9:00 | welcome and coffee<br /><br /><br />Morning session – chair: Jan SPRENGER (Tilburg)<br /><br /><br />9:30 | invited talk<br /><br /><b><a href="http://paulegre.free.fr/" target="_blank">Paul EGRÉ</a></b> (Institut Jean Nicod, Paris)<br /><br /><i>Logical consequence and ordinary reasoning</i><br /><br />The notion of logical consequence has been approached from a variety of angles. Tarski famously proposed a semantic characterization (in terms of truth-preservation), but also a structural characterization (in terms of axiomatic properties including reflexivity, transitivity, monotonicity, and other features). In recent work, E. Chemla, B. Spector and I have proposed a characterization of a wider class of consequence relations than Tarskian relations, which we call "respectable" (<i>Journal of Logic and Computation</i>, forthcoming). The class also includes non-reflexive and nontransitive relations, which can be motivated in relation to ordinary reasoning (such as reasoning with vague predicates, see Zardini 2008, Cobreros <i>et al</i>. 2012, or reasoning with presuppositions, see Strawson 1952, von Fintel 1998, Sharvit 2016). Chemla <i>et al</i>.'s characterization is partly structural, and partly semantic, however. In this talk I will present further advances toward a purely structural characterization of such respectable consequence relations. I will discuss the significance of this research program toward bringing logic closer to ordinary reasoning.<br /><br />(joint work with Emmanuel Chemla and Benjamin Spector)<br /><br /><br />10:30 | <b>Niels SKOVGAARD-OLSEN</b> (Freiburg)<br /><br /><i>Conditionals and multiple norm conflicts</i><br /><br /><br />11:15 | <b>Luis ROSA</b> (Munich)<br /><br /><i>Knowledge grounded on pure reasoning</i><br /><br /><br />12:00 | lunch break<br /><br /><br />Afternoon session I – chair: Steven HALES (Bloomsburg)<br /><br /><br />13:30 | invited talk<br /><br /><a href="http://lhenderson.org/" target="_blank"><b>Leah HENDERSON</b></a> (University of Groningen)<br /><br /><i>The unity of explanatory virtues</i><br /><br />Scientific theory choice is often characterised as an Inference to the Best Explanation (IBE) in which a number of distinct explanatory virtues are combined and traded off against one another. Furthermore, the epistemic significance of each explanatory virtue is often seen as highly case-specific. But are there really so many dimensions to theory choice? By considering how IBE may be situated in a Bayesian framework, I propose a more unified picture of the virtues in scientific theory choice.<br /><br /><br />14:30 | <b>Benjamin EVA</b> (Munich) and <b>Reuben STERN</b> (Munich)<br /><br /><i>Causal explanatory power</i><br /><br /><br />15:15 | coffee break<br /><br /><br />Afternoon session II – chair: Jakob KOSCHOLKE (Hamburg)<br /><br /><br />16:00 | <b>Barbara OSIMANI</b> (Munich)<br /><br /><i>Bias, random error, and the variety of evidence thesis</i><br /><br /><br />16:45 | <b>Felipe ROMERO</b> (Tilburg) and <b>Jan SPRENGER</b> (Tilburg)<br /><br /><i>Scientific self-correction: The Bayesian way</i><br /><br /><br /><br />ORGANIZING COMMITTEE<br /><br />Gustavo Cevolani (Torino)<br />Vincenzo Crupi (Torino)<br />Jason Konek (Kent)<br />Paolo Maffezioli (Torino)<br /><br /><br /><br />For any queries please contact Vincenzo Crupi (<a class="mailto" href="mailto:vincenzo.crupi@unito.it">vincenzo.crupi@unito.it</a><span class="mailto"></span>) or Jason Konek (<a class="mailto" href="mailto:jpkonek@ksu.edu">jpkonek@ksu.edu</a><span class="mailto"></span>).<br /><br /><br />Vincenzo Crupihttp://www.blogger.com/profile/08069145846190162517noreply@blogger.com0tag:blogger.com,1999:blog-4987609114415205593.post-43751540218688285602017-04-08T07:21:00.001+01:002017-04-08T07:22:37.836+01:00Formal Truth Theories workshop, Warsaw (Sep. 28-30)<div dir="ltr" style="text-align: left;" trbidi="on"><span style="text-align: justify;">Cezary Cieslinski and his team organize a workshop on formal theories of truth in Warsaw, to take place 28-30 September 2017. The invites include Dora Achourioti, Ali Enayat, Kentaro Fujimoto, Volker Halbach, Graham Leigh, and Albert Visser. Submission deadline is May 15. More details </span><a href="http://formaltruththeories.pl/call-for-papers/" style="text-align: justify;">here</a><span style="text-align: justify;">.</span></div>Rafal Urbaniakhttp://www.blogger.com/profile/10277466578023939272noreply@blogger.com0tag: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.com10tag: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.com0tag: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.com0