[FOM] multi-sorted logic

Richard Heck rgheck at brown.edu
Sun Mar 12 11:10:57 EST 2006

I could be wrong about this, but I don't know that there's all that much
to be said about multi-sorted logic (assuming we're in the classical
domain). Any multi-sorted theory can be made a single-sorted one by
introducing predicates S_1, .., S_n for the sorts and then relativizing
quantifiers. There will be things you do not preserve---for example,
philosophical interest---but proofs clearly will translate back and
forth. If one wants to do the model theory natively, then you just need
multiple domains, multiple styles of variables, multiple types of
predicates, and so forth. But then one just writes down the utterly
obvious semantics and proof-theory, and all the usual meta-theory
carries over.

Regarding higher-order logic, it'd be great if there were a "user's
guide", but I don't really know of one. Stewart Shapiro's book
/Foundations without Foundationalism/ contains some really useful
material, but it may be a bit too "encyclopedic". That said, Stewart's
discussion is directed more at applications than is, say, the discussion
in the later chapters of Boolos and Jeffrey. But if you put the two
together, that might work.


abaker1 at swarthmore.edu wrote:
> I recently received the following request from a colleague in the Mathematics 
> Department. Any suggestions would be welcome:
> "In my work recently I have been handicapped by not being familiar with
> the basic facts about mult-sorted and higher order logic.  Do you know a source 
> appropriate for a non-logician mathematician who wants something more on the 
> "user's guide" end of the spectrum rather than the encyclopedic end?"

Richard G Heck, Jr
Professor of Philosophy
Brown University
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