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