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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 tex2html_wrap_inline446 . Assume that reality is not directly observable. n may be very large, e.g. like Avogadro's number.

Let the values of the propositions tex2html_wrap_inline448 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 tex2html_wrap_inline450 , which are related to reality by


where l is also a modest number.

An approximate theory tex2html_wrap_inline519 is given by functions tex2html_wrap_inline452 , i.e. tex2html_wrap_inline519 undertakes to give what we want to know in terms of the observations.

If we are lucky in how reality turns out, the tex2html_wrap_inline454 functions correspond to the tex2html_wrap_inline456 functions, i.e.


for tex2html_wrap_inline458 , 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