Mercurial > dylan
changeset 10:543b1dbf821d
New article: Inductive lattices
author | Dylan Holmes <ocsenave@gmail.com> |
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date | Tue, 01 Nov 2011 01:55:26 -0500 |
parents | 23db8b1f0ee7 |
children | 1f112b4f9e8f |
files | org/visualizing-reason.org |
diffstat | 1 files changed, 104 insertions(+), 0 deletions(-) [+] |
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1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000 1.2 +++ b/org/visualizing-reason.org Tue Nov 01 01:55:26 2011 -0500 1.3 @@ -0,0 +1,104 @@ 1.4 +#+title:How to model Inductive Reasoning 1.5 +#+author: Dylan Holmes 1.6 +#+email: ocsenave@gmail.com 1.7 +##+description: An insight into plausible reasoning comes from experimenting with mathematical models. 1.8 +#+SETUPFILE: ../../aurellem/org/setup.org 1.9 +#+INCLUDE: ../../aurellem/org/level-0.org 1.10 + 1.11 +#Mathematics and computer science are the refineries of ideas. By 1.12 +#demanding unwavering precision and lucidness 1.13 + 1.14 +I've discovered a nifty mathematical presentation of 1.15 +plausible reasoning, which I've given the label *inductive posets* so 1.16 +that I can refer to the idea later. Though the idea of inductive posets has a number of shortcomings, it also 1.17 +shows some promise---there were a few resounding /clicks/ of agreement 1.18 +between the 1.19 +model and my intuition, and I got to see some exciting category-theoretic 1.20 +manifestations of some of my vaguer ideas. In this article, I'll talk about what I found particularly 1.21 +suggestive, and also what I found improvable. 1.22 + 1.23 + 1.24 +First, when you have a /deductive/ logical system, you can use a 1.25 +boolean lattice as a model. These boolean lattices capture ideas like 1.26 +deductive implication, negation, and identical truth/falsity. 1.27 + 1.28 +Suppose you have such a boolean lattice, \(L\), considered as a poset 1.29 +category with products defined between each of its members [fn::I haven't begun to think about big 1.30 +lattices, i.e. those with infinitely many atomic propositions. As 1.31 +such, let's consider just the finite case here.] and both an initial 1.32 +(\ldquo{}0\rdquo{}) and final (\ldquo{}1\rdquo{}) element. Now, using 1.33 +$L$ as a starting point, you can construct a new 1.34 +category $M$ as follows: the objects of $M$ are the same 1.35 +as the objects of $M$, and there is exactly one arrow 1.36 +\(A\rightarrow A\times B\) in $M$ for every pair of objects 1.37 +$A,B\in L$. 1.38 + 1.39 +Whereas we used $L$ to model deductive reasoning in a certain logical system, we will use 1.40 +this new lattice $M$ to model inductive reasoning in the same 1.41 +system. To do so, we will assign certain meanings to the features of 1.42 +$M$. Here is the key idea: 1.43 + 1.44 +#+begin_quote 1.45 +We'll interpret each arrow $A\rightarrow A\times B$ as the 1.46 + plausibility of $B$ given $A$. To strengthen the analogy, we'll 1.47 + sometimes borrow notation from probability theory, writing \((B|A)\) 1.48 + \(A\rightarrow A\times B\). 1.49 +#+end_quote 1.50 + 1.51 +This interpretation leads to some suggestive observations: 1.52 + 1.53 +- Certainty is represented by 1 :: You may know that the proposition \(A\Rightarrow B\) is logically 1.54 + equivalent to \(A=AB\). (If you haven't encountered this 1.55 + interesting fact yet, you should confirm it!) In our deductive 1.56 + lattice $L$, this equivalence means that there is an arrow $A\rightarrow B$ just if 1.57 + \(A\cong A\times B\) in \(L\). Relatedly, in our inductive lattice 1.58 + \(M\), this equivalence means that whenever $A\Rightarrow 1.59 + B$ in $L$, the arrow \(A\rightarrow A\times 1.60 + B\) is actually the (unique) arrow \(A\rightarrow A\). In 1.61 + probability theory notation, we write this as \((B|A)=1_A\) (!) This 1.62 + is a neat category-theoretic declaration of the usual 1.63 + result that the plausibility of a certainly true proposition is 1. 1.64 +- Deduction is included as a special case :: Because implications (arrows) in $L$ 1.65 + correspond to identity arrows in $M$, we have an inclusion 1.66 + functor \(\mathfrak{F}:L\rightarrow M\), which acts on arrows by 1.67 + sending \(A\rightarrow B\) to \(A\rightarrow A\times B\). This 1.68 +- Bayes' Law is a commutative diagram :: In his book on probability 1.69 + theory, Jaynes derives a product rule for plausibilities based 1.70 + 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 1.71 + states that \((AB|X) = (A|X)\cdot (B|AX) = (B|X)\cdot(A|BX)\). If 1.72 + we now work backwards to see what this statement in probability 1.73 + theory means in our inductive lattice \(M\), we find that it's 1.74 + astonishingly simple---Jaynes' product rule is just a commutative 1.75 + square: \((X\rightarrow ABX) = (X\rightarrow AX \rightarrow ABX) = 1.76 + (X\rightarrow BX\rightarrow ABX)\). 1.77 +- Inductive reasoning as uphill travel :: There is a certain analogy 1.78 + between the process of inductive reasoning and uphill travel: You 1.79 + begin in a particular state (your state of 1.80 + given information). From this starting point, you can choose to 1.81 + travel to other states. But travel is almost always uphill: to 1.82 + climb from a state of less information to a state of greater 1.83 + information incurs a cost in the form of low 1.84 + probability [fn::There are a number of reasons why I favor 1.85 + reciprocal probability---perhaps we could call it 1.86 + multiplicity?---and why I think reciprocal probability works 1.87 + better for category-theoretic approaches to probability 1.88 + theory. One of these is that, as you can see, reciprocal probabilities 1.89 + capture the idea of uphill costs. ]. Treating your newfound state 1.90 + as your new starting point, you can climb further. reaching states of successively higher information, while 1.91 + accumulating all the uphill costs. This analogy works well in a 1.92 + number of ways: it correctly shows that the probability of an 1.93 + event utterly depends on your current state of given information 1.94 + (the difficulty of a journey depends utterly on your starting 1.95 + point). It depicts deductive reasoning as zero-cost travel (the 1.96 + step from a proposition to one of its implications is /certain/ [fn::This is a thoroughly significant pun.] ---the travel is not 1.97 + precarious nor uphill, and there is no cost.) With the inductive 1.98 + lattice model in this article, we gain a new perspective of this 1.99 + travel metaphor: we can visualize inductive reasoning as the /accretion of given 1.100 + information/, going from \(X\rightarrow AX\rightarrow ABX\), and 1.101 + getting permission to use our current hypotheses as contingent 1.102 + givens by paying the uphill toll. 1.103 + 1.104 + 1.105 +# - The propositions are entirely syntactic; they lack internal 1.106 +# structure. This model has forgotten /why/ certain relations 1.107 +# hold. Possible repair is to