rlm@2: #+TITLE: Categorification of Plausible Reasoning
rlm@2: #+AUTHOR: Dylan Holmes
rlm@2: #+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
rlm@2: * COMMENT #+OPTIONS: LaTeX:dvipng
rlm@2: 
rlm@2: * Deductive and inductive posets
rlm@2: 
rlm@2: ** Definition
rlm@2: If you have a collection \(P\) of logical propositions, you can order them by
rlm@2: implication: \(a\) precedes \(b\) if and only if \(a\) implies
rlm@2: \(b\). This makes \(P\) into a poset. Since the ordering arose from
rlm@2: deductive implication, we'll call this a /deductive poset/.
rlm@2: 
rlm@2: If you have a deductive poset \(P\),  you can create a related poset \(P^*\)
rlm@2: as follows: the underlying set is the same, and for any two
rlm@2: propositions \(a\) and \(b\) in \(P\), \(a\) precedes
rlm@2: \(ab\) in \(P^*\). We'll call this an /inductive poset/.
rlm@2: 
rlm@2: ** A canonical map from  deductive posets to inductive posets
rlm@2: Each poset corresponds with a poset-category, that is a category with
rlm@2: at most one arrow between any two objects. Considered as categories,
rlm@2: inductive and deuctive posets are related as follows: there is a map
rlm@2: \(\mathscr{F}\) which sends each arrow \(a\rightarrow b\) in \(P\) to
rlm@2: the arrow \(a\rightarrow ab\) in \(P^*\). In fact, since \(a\) implies
rlm@2: \(b\) if and only if \(a = ab\), \(\mathscr{F}\) sends each arrow  in \(P\) to
rlm@2: an identity arrow  in \(P^*\) (specifically, it sends the arrow
rlm@2: \(a\rightarrow b\) to the identity arrow \(a\rightarrow a\)).
rlm@2: 
rlm@2: 
rlm@2: ** Assigning plausibilities to inductive posets
rlm@2: 
rlm@2: Inductive posets encode the relative (/qualitative/) plausibilities of its
rlm@2: propositions: there exists an arrow \(x\rightarrow y\) only if \(x\)
rlm@2: is at least as plausible as \(y\).
rlm@2: 
rlm@2: *** Consistent reasoning as a commutative diagram
rlm@2: Inductive categories enable the following neat trick: we can interpret
rlm@2: the objects of \(P^*\) as states of given information and interpret
rlm@2: each arrow \(a\rightarrow ab\) in \(P^*\) as an inductive inference: the arrow
rlm@2: \(a\rightarrow ab\) represents an inferential leap from the state of
rlm@2: knowledge where only \(a\) is given to the state of knowledge where
rlm@2: both \(a\) and \(b\) are given\mdash{} in this way, it represents
rlm@2: the process of inferring \(b\) when  given \(a\), and we label the
rlm@2: arrow with \((b|a)\).
rlm@2: 
rlm@2: This trick has several important features that suggest its usefulness,
rlm@2: namely
rlm@2:  - Composition of arrows corresponds to compound inference.
rlm@2:  - In the special case of deductive inference, the inferential arrow is an
rlm@2:    identity; the source and destination states of knowledge are the same.
rlm@2:  - One aspect of the consistency requirement of Jaynes[fn:1] takes the form of a
rlm@2:    commutative square: \(x\rightarrow ax \rightarrow abx\) =
rlm@2:    \(x\rightarrow bx \rightarrow abx\) is the categorified version of
rlm@2:    \((AB|X)=(A|X)\cdot(B|AX)=(B|X)\cdot(A|BX)\).
rlm@2:  - We can make plausibility assignments by enriching the inductive
rlm@2:    category \(P^*\) over some monoidal category, e.g. the set of real numbers
rlm@2:    (considered as a category) with its usual multiplication. /When we do/,
rlm@2:    the identity arrows of \(P^*\) \mdash{}corresponding to
rlm@2:    deductive inferences\mdash{} are assigned a value of certainty automatically.
rlm@2: 
rlm@2: [fn:1] /(IIIa) If a conclusion can be reasoned out in more than one
rlm@2: way, then every possible way must lead to the same result./
rlm@2: 
rlm@2: 
rlm@2: *** Reciprocal probabilities
rlm@2: The natural numbers have a comparatively concrete origin: they are the
rlm@2: result of decategorifying the category of finite sets[fn:2], or the
rlm@2: coequalizer of the arrows from a one-object category to a two-object
rlm@2: category with a single nonidentity arrow. Extensions of the set of
rlm@2: natural numbers\mdash{} such as
rlm@2: the set of integers or rational numbers or real numbers\mdash{} strike
rlm@2: me as being somewhat more abstract.
rlm@2: 
rlm@2: Jaynes points out that our existing choice of scale for probabilities
rlm@2: (i.e., the scale from 0 for impossibility to 1 for
rlm@2: certainty) has a degree of freedom: any monotonic function of
rlm@2: probability encodes the same information that probability does.
rlm@2: 
rlm@2: With this in mind, it seems useful to use not /probability/ but
rlm@2: /reciprocal probability/ instead. This scale, which we might
rlm@2: tentatively call freeness, is a scale ranging 1 (certainty) to
rlm@2: positive infinity (impossibility).
rlm@2: 
rlm@2: In this way, the ``probability''
rlm@2: associated with choosing one out of \(n\) indistinguishable choices
rlm@2: becomes identified with \(n\).
rlm@2: 
rlm@2: The entropy 
rlm@2: 
rlm@2: [fn:2] As Baez says.
rlm@2: 
rlm@2: 
rlm@2: 
rlm@2: ** self-questions
rlm@2: 
rlm@2: What circumstances would make \(\mathscr{F}\) an injection?
rlm@2: 
rlm@2: What if \(P=\{\top,\bot\}\)?
rlm@2: 
rlm@2: 
rlm@2: 
rlm@2: ** COMMENT 
rlm@2: Inductive and deductive posets are related as follows: there is a monotone
rlm@2: inclusion map \(\mathscr{i}:P^*\hookrightarrow P\) which\mdash{} since \(a\)
rlm@2: implies \(b\) is equivalent to \(a=ab\)\mdash{} sends comparable
rlm@2: propositions in \(P\) to the same proposition in \(P^*\). Conversely,
rlm@2: only comparable propositions in \(P\) are sent to the same proposition
rlm@2: in \(P^*\).
rlm@2: 
rlm@2: 
rlm@2: 
rlm@2: ** Inductive posets and plausibility
rlm@2: 
rlm@2: * Inverse Probability