> > But more seriously, HOW does one 'derive' axioms from a model?
>
>By induction (introducing universal quantifiers and implications),
>abduction (guessing, which may introduce any logical operators
>that seem "reasonable"), and further testing of any predictions
>derivable by deduction.
>
>This procedure provides a way to guarantee that the axioms are
>consistent: just check that every hypothesis derived by induction
>or abduction is consistent with the observational reports (i.e.,
>those that describe a suitable model). (01)
Typically, especially in recent years, the hypotheses are
statistical. Furthermore, the issue typically is not whether the
axioms of the theory are consistent with the data. The logical
constraints in statistical theories are extremely weak, admitting
wildly improbable data. (E.g., a sequence of 20,000 tosses of a fair
coin, all coming up heads, is consistent with the logical axioms of
the statistical theory that the coin is fair.) In a good statistical
theory, the data will not be EXACTLY consistent with the statistical
axioms. That is, the actual frequencies will not conform to the
predicted frequencies under the statistical theory. (For example, we
would not expect EXACTLY 5000 out of 10,000 tosses of a fair coin to
come up heads. In fact, people think Mendel fudged his data because
the frequencies are too good.) Statistical tests evaluate the degree
to which the observations are a "typical" realization of the
hypothesized data-generating process. (02)
K (03)
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