Basic Desiderata

In order to focus on constructive reasoning and avoid irrelevancies, we invent a third party, called "robot", that will use simple rules to reason about propositions. These rules are the basic desiderata explained in this section.

The propositions are denoted by capital letters $A, B, C, \ldots$ and must have an unambiguous objective meaning of simple, definite logical type that must be either true or false.

The robot must assign a degree of plausibility to each proposition based on the evidence and update assignments in light of new evidence. These plausibility assignments have to be represented as some quantity, and we choose real numbers. Also, a greater plausibility shall correspond to a greater number, and an infinitesimally greater plausibility has to correspond to an infinitesimally greater number.

(I) Degrees of plausibility are represented by real numbers.

The plausibility that the robot assigns to some proposition $A$ will in general depend on whether we told it that some other proposition $B$ is true, which we denote by the following symbol:

$$A \, | \, B, \tag{1.29}$$

which we call the conditional plausibility that $A$ is true, given that $B$ is true or just "A given B", which stands for a real number.

$$A+B \, | \, CD \tag{1.31}$$

represents the plausibility that at least one of the propositions $A$ and $B$ is true, given that both $C$ and $D$ are true.

Because we have decided that a greater plausibility is represented by a greater number,

$$ (A \, | \, B) \gt (C \, | \, B) \tag{1.32}$$

says that given $B$, $A$ is more plausible than $C$. We can also express it without parentheses as

$$ A \, | \, B \gt C \, | \, B \tag{1.33}$$

We avoid impossible problems of defining $A \, | \, BC$ when $B$ and $C$ are mutually contradictory by assuming that $B$ and $C$ are compatible propositions.

If old information $C$ is updated to $C'$ so that the plausibility for $A$ is increased:

$$A \, | \, C' \gt A \, | \, C$$

but the plausibility for $B$ given $A$ is not changed:

$$B \, | \, AC' = B \, | \, AC$$

then the plausibility that both $A$ and $B$ are true can only increase, never decrease:

$$AB \, | \, C' \ge AB \, | \,C$$

and it must produce a decrease in the plausibility that $A$ is false:

$$\overline A \, | \, C' \lt \overline A \, | \, C$$

A greater plausibility shall correspond to a greater number, and an infinitesimally greater plausibility have to correspond to an infinitesimally greater number (continuity).

(II) Qualitative correspondence with common sense.

Finally, we want to give our robot another desirable property that is always reasons consistently:

(IIIa) If a colcusion can be reasoned out in more than one way, then every possible way must lead to the same result.

(IIIb) The robot always takes into account all of the evidence it has relevant to a question. It does not arbitrarily ignore some of the information, basing its conclusions only on what remains. In other words, the robot is completely nonideological.

(IIIc) The robot always represents equivalent states of knowledge by equivalent plausibility assignments. That is, if in two problems the robot's state of knowledge is the same (except perhaps for the labeling of the propositions), then it must assign the same plausibilities in both.

Desiderata (I), (II), and (IIIa) are the basic structural requirements on the inner workings of the robots's brain, while (IIIb) and (IIIc) are interface conditions which show how the robot's behavior should relate to the outer world.

The above conditions uniquely determine the rules of reasoning; there is only one set of mathematical operations for manipulating plausibilities.