Pseudo-quantification in Trade Logic

Pseudo-quantification in Trade Logic via best

Previously

Earlier I introduced Trade Logic. It's a form of logic that I designed to be suitable for connecting prediction markets. It is logic that "lives" within the system, not in individual traders' analyses.

Not quantification

Trade Logic doesn't use quantification as such. Quantification would complicate the mode system, which is already adequate to distinguish free and bound variables. I also have theoretical concerns about vacuous quantification: If there were no objects that the system could refer to, the relative valuation of (forall x (p x)) and (some x (p x)) would be reversed.

The built-in predicate best

Instead of directly using quantification, Trade Logic uses the built-in predicate best, which is adapted from Hilbert's epsilon operator. best(A,B) is true just if:

• B is a unary predicate
• A satisfies B as well as anything else does

I'll expand a little on that last point. That's not the same as "satisfies B". best can be true if no value could satisfy B. In Trade Logic, best can also be true in fuzzy ways:

• If B can only be satisfied to a certain degree, and A satisfies B to that degree, then best(A,B) is true (crisply, 100%)
• if A satisfies B to a certain degree, but a lesser degree than some other value would, then best(A,B) is fuzzily true

Quantifiers can be expressed in terms of best, as they could with epsilon. In standard notation, we would write:

\begin{equation} \forall x p(x) \Leftrightarrow best(x, \neg p(x)) \rightarrow p(x) \end{equation} \begin{equation} \exists x p(x) \Leftrightarrow best(x, p(x)) \rightarrow p(x) \end{equation}

In Trade Logic, the respective formulas are:

(if (best +x (lambda (Y) (not (-p Y)))) (-p -x))

and

(if (best +x -p) (-p -x)) 

Note the addition of modes, and note that "p" is always an in mode. It must be bound outside this (sub)formula.

The behavior of best

A yes of any issue of the form (best +A -B) can be converted to a yes of (& (-X +A) (-B -A)) for any predicate X. Similarly, a no of (& (-X +A) (-B -A)) can be converted to a no of (best +A -B).

X may select A in an arbitrary way, but it will never be better at satisfying B than (best +A -B) is.

This works because no trader would make this conversion unless he got a better price after the conversion. This ensures that the price of best issues is always in fact the highest price. Effectively, existentially quantified issues are always as high or higher in price than each of their particular instances, and universally quantified issues as low or lower than their particular instances.

Best is adapted from epsilon

Best is adapted from Hilbert's epsilon operator. (Also see here) Epsilon (not best) classically has the following properties:

• It is a function
  • Usually it comes with an axiom of extensionality, ie that the function's result is unique.
• It takes one argument, a predicate (as a formula)
• If that predicate can be satisfied, it returns an object that satisfies the predicate
• If that predicate can't be satisfied, it returns any object at all.
• With it, one can build statements equivalent to other statements that contain universal and existential quantifiers.
• The quantifiers all and some can be expressed using it.

\begin{equation} \forall x p(x) \Leftrightarrow p(\varepsilon(x, \neg p(x))) \end{equation} \begin{equation} \exists x p(x) \Leftrightarrow p(\varepsilon(x, p(x))) \end{equation}

But Trade Logic doesn't contain functions. What Trade Logic has are fuzzy predicates¹. So we use best instead.

Footnotes:

¹ When I first saw Epsilon, I got the impression that it wasn't suitable for this reason. But I was wrong, it just needed to be adapted.