Reason and Risk: An Account of Induction

We begin with a non-empty, finite set of propositions that have already been accepted and compiled and now reside in a database of accepted propositions (DAP). Assume there is some known risk (possible zero) associated with accepting these propositions. Let the level of risk of the nth proposition Pn be a non-negative number, R(Pn), which is recorded in the DAP.

The question is, Given this DAP, what other propositions might be accepted with no increase in risk? Certainly there are enumerative propositions about the DAP, such as All x’s are y’s in the DAP, or If x is a y, then x is a z in the DAP. Such enumeration is pre-inductive but sets the stage for the inductive step.

Some pre-inductive propositions may contain terms that function as fields in a database. Whether the DAP is actually structured this way is not of consequence here. Some of these fields may have references beyond the DAP. For example, if the DAP contains propositions about employees, then it is possible that there are employees that exist but are not included in the DAP.

The point is that enumerative propositions that are true about the DAP might be true if the DAP were enlarged to contain more propositions. It is acknowledged that there would be risk associated with accepting a proposition that refers beyond the DAP, but that risk can be accounted for and ameliorated to some degree.

The approach is ameliorating risk is the natural kind. A natural kind is a set of things which are homomorphic with a proper subset of itself. An example would be a natural kind such as copper in which the properties of one sample of copper are homomorphic with those of another sample.

There is a risk as to whether or not something in the DAP is actually a member of a natural kind or which kind it is a member of. There is also a risk whether or not a particular property of the natural kind is one of the properties that is natural.

These risks are ameliorated by two strategies: (1) increasing the measure of risk associated with a proposition that refers beyond the DAP so as to capture this risk lest anyone be misled, and (2) searching for limits to the natural kind, both as to what properties are held throughout the kind and as to what the limits of the kind itself are.

For example, we might find in the DAP that All employees are female in the DAP. We may consider females a natural kind so that other females have like properties and infer that All employees are female, whether they are mentioned in the DAP or not, but we would have to increase the risk of this statement and search for limits to the set of female employees.

Until the limits to a proposition are found, its risk is higher than a proposition only about the DAP. If, for example, an employee outside the DAP is found to be male, then a limit to the inferred proposition has been found. In that case, the inferred proposition should be restricted to remain within the limit, and the risk lowered.

So if a set of things is proposed as a natural kind, then there is some risk whether or not it is in actuality the proposed natural kind. There is also the risk whether or not a property of a sample of the kind is a property of the rest of the kind. Everything in the kind must have some properties in common but there will always be some properties that are unique to each sample, otherwise the sample could not even be distinguished.

Looking at this situation globally we may say that every entity is a member of some natural kind. After all, there can be only one truly unique entity in the universe, that is, the universal entity. All other entities have properties in common with other entities, and to that extent they are members of the same natural kind. The question then becomes, Which other properties do they have in common?

January 2015