Mercurial > dylan
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New article: Inductive lattices
| author | Dylan Holmes <ocsenave@gmail.com> |
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| date | Tue, 01 Nov 2011 01:55:26 -0500 |
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| children | 1f112b4f9e8f |
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| 1 #+title:How to model Inductive Reasoning | |
| 2 #+author: Dylan Holmes | |
| 3 #+email: ocsenave@gmail.com | |
| 4 ##+description: An insight into plausible reasoning comes from experimenting with mathematical models. | |
| 5 #+SETUPFILE: ../../aurellem/org/setup.org | |
| 6 #+INCLUDE: ../../aurellem/org/level-0.org | |
| 7 | |
| 8 #Mathematics and computer science are the refineries of ideas. By | |
| 9 #demanding unwavering precision and lucidness | |
| 10 | |
| 11 I've discovered a nifty mathematical presentation of | |
| 12 plausible reasoning, which I've given the label *inductive posets* so | |
| 13 that I can refer to the idea later. Though the idea of inductive posets has a number of shortcomings, it also | |
| 14 shows some promise---there were a few resounding /clicks/ of agreement | |
| 15 between the | |
| 16 model and my intuition, and I got to see some exciting category-theoretic | |
| 17 manifestations of some of my vaguer ideas. In this article, I'll talk about what I found particularly | |
| 18 suggestive, and also what I found improvable. | |
| 19 | |
| 20 | |
| 21 First, when you have a /deductive/ logical system, you can use a | |
| 22 boolean lattice as a model. These boolean lattices capture ideas like | |
| 23 deductive implication, negation, and identical truth/falsity. | |
| 24 | |
| 25 Suppose you have such a boolean lattice, \(L\), considered as a poset | |
| 26 category with products defined between each of its members [fn::I haven't begun to think about big | |
| 27 lattices, i.e. those with infinitely many atomic propositions. As | |
| 28 such, let's consider just the finite case here.] and both an initial | |
| 29 (\ldquo{}0\rdquo{}) and final (\ldquo{}1\rdquo{}) element. Now, using | |
| 30 $L$ as a starting point, you can construct a new | |
| 31 category $M$ as follows: the objects of $M$ are the same | |
| 32 as the objects of $M$, and there is exactly one arrow | |
| 33 \(A\rightarrow A\times B\) in $M$ for every pair of objects | |
| 34 $A,B\in L$. | |
| 35 | |
| 36 Whereas we used $L$ to model deductive reasoning in a certain logical system, we will use | |
| 37 this new lattice $M$ to model inductive reasoning in the same | |
| 38 system. To do so, we will assign certain meanings to the features of | |
| 39 $M$. Here is the key idea: | |
| 40 | |
| 41 #+begin_quote | |
| 42 We'll interpret each arrow $A\rightarrow A\times B$ as the | |
| 43 plausibility of $B$ given $A$. To strengthen the analogy, we'll | |
| 44 sometimes borrow notation from probability theory, writing \((B|A)\) | |
| 45 \(A\rightarrow A\times B\). | |
| 46 #+end_quote | |
| 47 | |
| 48 This interpretation leads to some suggestive observations: | |
| 49 | |
| 50 - Certainty is represented by 1 :: You may know that the proposition \(A\Rightarrow B\) is logically | |
| 51 equivalent to \(A=AB\). (If you haven't encountered this | |
| 52 interesting fact yet, you should confirm it!) In our deductive | |
| 53 lattice $L$, this equivalence means that there is an arrow $A\rightarrow B$ just if | |
| 54 \(A\cong A\times B\) in \(L\). Relatedly, in our inductive lattice | |
| 55 \(M\), this equivalence means that whenever $A\Rightarrow | |
| 56 B$ in $L$, the arrow \(A\rightarrow A\times | |
| 57 B\) is actually the (unique) arrow \(A\rightarrow A\). In | |
| 58 probability theory notation, we write this as \((B|A)=1_A\) (!) This | |
| 59 is a neat category-theoretic declaration of the usual | |
| 60 result that the plausibility of a certainly true proposition is 1. | |
| 61 - Deduction is included as a special case :: Because implications (arrows) in $L$ | |
| 62 correspond to identity arrows in $M$, we have an inclusion | |
| 63 functor \(\mathfrak{F}:L\rightarrow M\), which acts on arrows by | |
| 64 sending \(A\rightarrow B\) to \(A\rightarrow A\times B\). This | |
| 65 - Bayes' Law is a commutative diagram :: In his book on probability | |
| 66 theory, Jaynes derives a product rule for plausibilities based | |
| 67 on his [[http://books.google.com/books?id=tTN4HuUNXjgC&lpg=PP1&dq=Jaynes%20probability%20theory&pg=PA19#v=onepage&q&f=fals][criterion for consistent reasoning]]. This product rule | |
| 68 states that \((AB|X) = (A|X)\cdot (B|AX) = (B|X)\cdot(A|BX)\). If | |
| 69 we now work backwards to see what this statement in probability | |
| 70 theory means in our inductive lattice \(M\), we find that it's | |
| 71 astonishingly simple---Jaynes' product rule is just a commutative | |
| 72 square: \((X\rightarrow ABX) = (X\rightarrow AX \rightarrow ABX) = | |
| 73 (X\rightarrow BX\rightarrow ABX)\). | |
| 74 - Inductive reasoning as uphill travel :: There is a certain analogy | |
| 75 between the process of inductive reasoning and uphill travel: You | |
| 76 begin in a particular state (your state of | |
| 77 given information). From this starting point, you can choose to | |
| 78 travel to other states. But travel is almost always uphill: to | |
| 79 climb from a state of less information to a state of greater | |
| 80 information incurs a cost in the form of low | |
| 81 probability [fn::There are a number of reasons why I favor | |
| 82 reciprocal probability---perhaps we could call it | |
| 83 multiplicity?---and why I think reciprocal probability works | |
| 84 better for category-theoretic approaches to probability | |
| 85 theory. One of these is that, as you can see, reciprocal probabilities | |
| 86 capture the idea of uphill costs. ]. Treating your newfound state | |
| 87 as your new starting point, you can climb further. reaching states of successively higher information, while | |
| 88 accumulating all the uphill costs. This analogy works well in a | |
| 89 number of ways: it correctly shows that the probability of an | |
| 90 event utterly depends on your current state of given information | |
| 91 (the difficulty of a journey depends utterly on your starting | |
| 92 point). It depicts deductive reasoning as zero-cost travel (the | |
| 93 step from a proposition to one of its implications is /certain/ [fn::This is a thoroughly significant pun.] ---the travel is not | |
| 94 precarious nor uphill, and there is no cost.) With the inductive | |
| 95 lattice model in this article, we gain a new perspective of this | |
| 96 travel metaphor: we can visualize inductive reasoning as the /accretion of given | |
| 97 information/, going from \(X\rightarrow AX\rightarrow ABX\), and | |
| 98 getting permission to use our current hypotheses as contingent | |
| 99 givens by paying the uphill toll. | |
| 100 | |
| 101 | |
| 102 # - The propositions are entirely syntactic; they lack internal | |
| 103 # structure. This model has forgotten /why/ certain relations | |
| 104 # hold. Possible repair is to |