annotate org/visualizing-reason.org @ 10:543b1dbf821d

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