FOM: Second order logic
John Mayberry
J.P.Mayberry at bristol.ac.uk
Thu Mar 11 11:28:35 EST 1999
In the controversy over second order logic, everyone seems to
agree about the basic mathematical facts: the disagreement seems to be
over how the facts should be viewed and how we should describe them.
According to Steve Simpson, second order logic (with the standard
semantics) is not logic because "it doesn't provide a model of
reasoning". But then first order logic doesn't provide a model of
reasoning either: it can't capture the reasoning we use when we employ
the axiomatic method to define the various sorts of mathematical
structure: it works for some such definitions (e.g. the definition of
"group") but not for others (e.g. the definition of "complete ordered
field" or "topological space").
I can anticipate an objection: "these axiomatic definitions
belong to set theory, and set theory is not part of logic". I agree.
Set theory is not part of logic: logic - formal, mathematical logic -
is part of set theory. Without set theory you cannot motivate your
systems of formal, first order proof, because you can't formulate
completeness in a natural way, or establish it.
John Mayberry
School of Mathematics
University of Bristol
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John Mayberry
J.P.Mayberry at bristol.ac.uk
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