Logical Inductor Notation and Definitions


by Alex Appel Apr 12 2017 updated Apr 21 2017

A handy guide to the blizzard of notation in the Logical Induction Paper

It should be noted that, although there are many new definitions introduced in the Logical Induction (incomplete) paper, there is a great deal of internal structure that recurs over and over, like the Standard Model of physics. This page will try to reveal the underlying structure that generates the definitions, as well as what the definitions are used for, because processing all the new definitions is essential for truly understanding the paper.

There are four general "conceptual slots" to establish, and a substantial list of minor modifications to each of the slots.

The following image is useful to blow up and keep as a reference for reading the paper.


Anyways, the first important "conceptual slot" is the "sentence" slot. These can be interpreted as sentences of math, or as stocks that the "traders" are trading. $~$\mathcal{S}$~$ is the set of math sentences.

The second "conceptual slot" is the "valuation" slot. This assigns stuff from the "sentence" slot a particular value, that is typically interpreted as a probability/price. They are generally functions $~$\mathbb{V}:\mathcal{S}\rightarrow\mathbb{R}$~$

The third "conceptual slot" is the "coefficient" slot. You could think of this as the function a particular trader uses to determine how many shares of a particular sentence to buy. They rely on the history of past valuations, and have type signature $~$\alpha:\overline{\mathbb{V}}\rightarrow\mathbb{R}$~$

The fourth "conceptual slot" is the "combination" slot. This is the actual trade a "trader" makes on a particular day, and is represented as a linear combination of stuff from the "sentence" slot, with the coefficients being stuff from the "coefficient" slot. The coefficient for a particular sentence, when run on the past "valuation history" $~$\overline{\mathbb{V}}$~$, produces the number of shares of that sentence bought or sold on that day. It is possible to abuse notation to get the "valuations" to value "combinations" instead of sentences.

There are many additional subtleties, but these are the basic categories to become fluent with. Feel free to refer back to this page as needed.

Sentence Slot

Deductive Process

Valuation Slot

Coefficient Slot

Combination Slot

Valuations of Combinations

Expectation Operators

Indicator Functions

Number Sequences

(conditional definitions omitted since that proof still needs to be fixed)

(also still need to put the actual definition of a logical inductor)