## 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 φ is a formula with free occurrences of exactly the variables
*v*_{1}, ..., *v*_{n}, then
‘[λ*v*_{1},...,
*v*_{n} φ]’ 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 φ to contain no free variables (in
which case ‘[λ φ]’ denotes the proposition
*that*-φ) or more free variables than are bound by the
λ-operator (to allow expressions like
‘[λ*x* *Fxyz*]’, which we could
quantify into, as with
‘∃*y*([λ*x* *Fxyz*])’.

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