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Notes to Properties

1. Let
**I** be an interpretation function with respect to a
model, so that **I**(*a*) is the individual
**I** assigns to the individual constant
‘*a*’,
**I**(*F*^{n}) is the
*n*-place relation that **I** assigns to
*n*-place predicate *F*^{n}. Then for
all *n*-place predicates
‘*F*^{n}’ and individuals constants
‘*c*_{1}’, ...,
‘*c*_{n}’, the sentence
‘*F*^{n}*c*_{1}...*c*_{n}’
is true in the interpretation just in case <
**I**(*c*_{1}), ...,
**I**(*c*_{n})>
**ext**(**I**(*F*^{n})). We
would relativize this clause to variable assignments in the usual way
to define satisfaction conditions for atomic sentences.

2. More
generally, if
is a formula with free occurrences of exactly the
variables *v*_{1}, ..., *v*_{n}, then
‘[*v*_{1},...,
*v*_{n} ]’
is an *n*-place complex predicate (normal quotation marks are
used here as stand-ins for quasi-quotation). In more complicated
applications we could allow to contain no free
variables (in which case
‘[ ]’ denotes the
proposition *that*-) or more free variables
than are bound by the -operator (to allow
expressions like
‘[*x* *Fxyz*]’, which we could
quantify into, as with
‘*y*([*x* *Fxyz*])’.

3. A standard sort of
comprehension schema
holds that for each open formula
,
*X*^{n}*x*_{1}...*x*_{n}(*X*^{n}*x*_{1}...*x*_{n}
if and only if ).

Chris Swoyer

*First published: September 23, 1999*

*Content last modified: September 23, 1999*