#+title:A Category-Theoretic View of Inductive Reasoning

 #+author: Dylan Holmes

 #+email: ocsenave@gmail.com

 ##+description: An insight into plausible reasoning comes from experimenting with mathematical models. 

 #+SETUPFILE: ../../aurellem/org/setup.org

 #+INCLUDE: ../../aurellem/org/level-0.org

 

 #Mathematics and computer science are the refineries of ideas. By

 #demanding unwavering precision and lucidness  

 

 I've discovered a nifty mathematical presentation of

 plausible reasoning, which I've given the label *inductive posets* so

 that I can refer to the idea later. Though the idea of inductive posets has a number of shortcomings, it also

 shows some promise---there were a few resounding /clicks/ of agreement

 between the

 model and my intuition, and I got to see some exciting category-theoretic 

 manifestations of some of my vaguer ideas. In this article, I'll talk about what I found particularly

 suggestive, and also what I found improvable.

 

 

 First, when you have a /deductive/ logical system, you can use a

 boolean lattice as a model. These boolean lattices capture ideas like

 deductive implication, negation, and identical truth/falsity.

 

 Suppose you have such a boolean lattice, \(L\), considered as a poset

 category with products defined between each of its members [fn::I haven't begun to think about big

 lattices, i.e. those with infinitely many atomic propositions. As

 such, let's consider just the finite case here.] and both an initial

 (\ldquo{}0\rdquo{}) and final (\ldquo{}1\rdquo{}) element. Now, using

 $L$ as a starting point, you can construct a new

 category $M$ as follows: the objects of $M$ are the same

 as the objects of $M$, and there is exactly one arrow

 \(A\rightarrow A\times B\) in $M$ for every pair of objects

 $A,B\in L$. 

 

 Whereas we used $L$ to model deductive reasoning in a certain logical system, we will use

 this new lattice $M$ to model inductive reasoning in the same

 system. To do so, we will assign certain meanings to the features of

 $M$. Here is the key idea:

 

 #+begin_quote

 We'll interpret each arrow $A\rightarrow A\times B$ as the

   plausibility of $B$ given $A$. To strengthen the analogy, we'll

   sometimes borrow notation from probability theory, writing \((B|A)\)

   \(A\rightarrow A\times B\). 

 #+end_quote

 

 This interpretation leads to some suggestive observations:

 

 - Certainty is represented by 1 :: You may know that the proposition \(A\Rightarrow B\) is logically

   equivalent to \(A=AB\). (If you haven't encountered this

   interesting fact yet, you should confirm it!) In our deductive

      lattice $L$, this equivalence means that there is an arrow $A\rightarrow B$ just if

      \(A\cong A\times B\) in \(L\). Relatedly, in our inductive lattice

      \(M\), this equivalence means that whenever $A\Rightarrow

      B$ in $L$, the arrow \(A\rightarrow A\times

      B\) is actually the (unique) arrow \(A\rightarrow A\). In

      probability theory notation, we write this as \((B|A)=1_A\) (!) This

      is a neat category-theoretic declaration of the usual

      result that the plausibility of a certainly true proposition is 1.

 - Deduction is included as a special case :: Because implications (arrows) in $L$

      correspond to identity arrows in $M$, we have an inclusion

      functor \(\mathfrak{F}:L\rightarrow M\), which acts on arrows by

      sending \(A\rightarrow B\) to \(A\rightarrow A\times B\). This

 - Bayes' Law is a commutative diagram ::  In his book on probability

      theory, Jaynes derives a product rule for plausibilities based

      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

      states that \((AB|X) = (A|X)\cdot (B|AX) = (B|X)\cdot(A|BX)\). If

      we now work backwards to see what this statement in probability 

      theory means in our inductive lattice \(M\), we find that it's

      astonishingly simple---Jaynes' product rule is just a commutative

      square: \((X\rightarrow ABX) = (X\rightarrow AX \rightarrow ABX) =

      (X\rightarrow BX\rightarrow ABX)\). 

 - Inductive reasoning as uphill travel :: There is a certain analogy

      between the process of inductive reasoning and uphill travel: You

      begin in a particular state (your state of

      given information). From this starting point, you can choose to

      travel to other states. But travel is almost always uphill: to

      climb from a state of less information to a state of greater

      information incurs a cost in the form of low

      probability [fn::There are a number of reasons why I favor

      reciprocal probability---perhaps we could call it

      multiplicity?---and why I think reciprocal probability works

      better for category-theoretic approaches to probability

      theory. One of these is that, as you can see, reciprocal probabilities 

      capture the idea of uphill costs. ]. Treating your newfound state

      as your new starting point, you can climb further. reaching states of successively higher information, while

      accumulating all the uphill costs. This analogy works well in a

      number of ways: it correctly shows that the probability of an

      event utterly depends on your current state of given information

      (the difficulty of a journey depends utterly on your starting

      point). It depicts deductive reasoning as zero-cost travel (the

      step from a proposition to one of its implications is /certain/ [fn::This is a thoroughly significant pun.] ---the travel is not

      precarious nor uphill, and there is no cost.)  With the inductive

      lattice model in this article, we gain a new perspective of this

      travel metaphor: we can visualize inductive reasoning as the /accretion of given

      information/, going from \(X\rightarrow AX\rightarrow ABX\), and

      getting permission to use our current hypotheses as contingent

      givens by paying the uphill toll.

       

 

 # - The propositions are entirely syntactic; they lack internal

 #  structure. This model has forgotten /why/ certain relations

 #  hold. Possible repair is to