### Progic II

#### Posted by David Corfield

My research time at present largely consists of the time it takes me to travel to Canterbury and back. Here’s what I came up with on the subject of progic on yesterday’s trip down.

A probabilistic predicate, $P$, I said back here, is a map from $X$ to $[0, 1]$. But another way to look at it is as a conditional probability distribution on the set $2$ given $X$. This is a map in the Kleisli category of the probability monad. In fact, our probabilistic predicate is a map $P: X \to 2$. Now we can look at the composition of a distribution over $X$ and this $P$.

Let’s take $X$ a set of dogs, $Q(x)$ is a probability distibution over $X$, perhaps recording my degree of belief in which dog I saw just now as it dashed past. Then if $P(x)$ is the predicate recording my belief as to whether $x$ is a poodle, then $\sum_x P(x) \times Q(x)$ is my degree of belief that I just glimpsed a poodle.

This is one of Mumford’s random constants, i.e., probabilistic predicates over {*}, or maps in the Kleisli category $1 \to 2$.

Now how much of all that stuff about functions between $X$ and $Y$ and adjunctions that I mentioned before can we bring with us? First, why not have *probabilistic* functions, or in other words conditional distributions $Own(y | x)$, e.g., recording my uncertainty as to the owner of dog $x$? Then for every probabilistic predicate on humans, I can pull it back to give a probabilistic predicate on dogs. E.g., if $F(y)$ records my degree of belief that person $y$ is French, I have a degree of belief, $\sum_y F(y) \cdot Own(y | x)$ that $x$ is owned by a Frenchman, and $\sum_{x, y} F(y) \cdot Own(y | x) \cdot Q(x)$ that the dog I just glimpsed is owned by a Frenchman.

But can we form those adjunctions? An obvious choice of a structure to place on the probabilistic predicates on $X$ is a partial order where $P \leq Q$ if and only if $P(x) \leq Q(x)$, for all $x \in X$. So if ‘Poodle’ $\leq$ ‘owned by a Frenchman’, do we have something like ‘owner of at least one poodle’ $\leq$ ‘French’? It seems to me unlikely that there are mappings from probabilistic predicates over $Y$ to those over $X$, adjoint to this ‘multiplication’ by $Own(y | x)$.

## Re: Progic II

We can see the adjoints at play in the case of predicate logic preserving (co)limits.

Substitution being both a left and right adjoint, we have

(There being just one owner.)

So, the product and coproduct are preserved.

Now for the adjoints:

But now ‘and’, including the one implicit in ‘black poodles’, cannot be replaced by ‘or’.

Now ‘or’ cannot be replaced by ‘and’.

This nice picture is going to fall apart presumably in the probabilistic case.