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# Propositional approximate theories

Many topics take an especially simple form when one uses propositions instead of predicates--and accepts the reduced expressivity.

Here is one approach to defining approximate propositional theories.

Let reality, e.g. the situation in a room, be given by the values of the propositional variables . Assume that reality is not directly observable. n may be very large, e.g. like Avogadro's number.

Let the values of the propositions be observable. They are functions of reality given by

where k is a modest number corresponding to how many bits we can actually observe.

We suppose that we want to know the values of , which are related to reality by

where l is also a modest number.

An approximate theory is given by functions , i.e. undertakes to give what we want to know in terms of the observations.

If we are lucky in how reality turns out, the functions correspond to the functions, i.e.

for , i.e.

If we are very fortunate we may be able to know when we are lucky, and we have

At the moment, we have no useful propositional approximate theories in mind, and the reader should remember Einstein's dictum ``Everything should be made as simple as possible--but not simpler.''

John McCarthy
Wed Feb 2 15:59:04 PST 2000