FOM: Second order logic

[John Mayberry]

	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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